Documentation

Mathlib.LinearAlgebra.LinearPMap

Partially defined linear maps #

A LinearPMap R E F or E →ₗ.[R] F is a linear map from a submodule of E to F. We define a SemilatticeInf with OrderBot instance on this, and define three operations:

Moreover, we define

Partially defined maps are currently used in Mathlib to prove Hahn-Banach theorem and its variations. Namely, LinearPMap.sSup implies that every chain of LinearPMaps is bounded above. They are also the basis for the theory of unbounded operators.

structure LinearPMap (R : Type u) [Ring R] (E : Type v) [AddCommGroup E] [Module R E] (F : Type w) [AddCommGroup F] [Module R F] :
Type (max v w)

A LinearPMap R E F or E →ₗ.[R] F is a linear map from a submodule of E to F.

Instances For

    A LinearPMap R E F or E →ₗ.[R] F is a linear map from a submodule of E to F.

    Equations
    • One or more equations did not get rendered due to their size.
    Instances For
      def LinearPMap.toFun' {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
      f.domainF
      Equations
      • f = f.toFun
      Instances For
        Equations
        • LinearPMap.instCoeFunLinearPMapForAllSubtypeMemSubmoduleToSemiringToAddCommMonoidInstMembershipSetLikeDomain = { coe := LinearPMap.toFun' }
        @[simp]
        theorem LinearPMap.toFun_eq_coe {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : f.domain) :
        f.toFun x = f x
        theorem LinearPMap.ext {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (h : f.domain = g.domain) (h' : ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄, x = yf x = g y) :
        f = g
        @[simp]
        theorem LinearPMap.map_zero {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
        f 0 = 0
        theorem LinearPMap.ext_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} :
        f = g ∃ (_ : f.domain = g.domain), ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄, x = yf x = g y
        theorem LinearPMap.ext' {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {s : Submodule R E} {f : s →ₗ[R] F} {g : s →ₗ[R] F} (h : f = g) :
        { domain := s, toFun := f } = { domain := s, toFun := g }
        theorem LinearPMap.map_add {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : f.domain) (y : f.domain) :
        f (x + y) = f x + f y
        theorem LinearPMap.map_neg {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : f.domain) :
        f (-x) = -f x
        theorem LinearPMap.map_sub {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : f.domain) (y : f.domain) :
        f (x - y) = f x - f y
        theorem LinearPMap.map_smul {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (c : R) (x : f.domain) :
        f (c x) = c f x
        @[simp]
        theorem LinearPMap.mk_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (f : p →ₗ[R] F) (x : p) :
        { domain := p, toFun := f } x = f x
        noncomputable def LinearPMap.mkSpanSingleton' {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c x = 0c y = 0) :

        The unique LinearPMap on R ∙ x that sends x to y. This version works for modules over rings, and requires a proof of ∀ c, c • x = 0 → c • y = 0.

        Equations
        • One or more equations did not get rendered due to their size.
        Instances For
          @[simp]
          theorem LinearPMap.domain_mkSpanSingleton {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c x = 0c y = 0) :
          @[simp]
          theorem LinearPMap.mkSpanSingleton'_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c x = 0c y = 0) (c : R) (h : c x (LinearPMap.mkSpanSingleton' x y H).domain) :
          (LinearPMap.mkSpanSingleton' x y H) { val := c x, property := h } = c y
          @[simp]
          theorem LinearPMap.mkSpanSingleton'_apply_self {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : E) (y : F) (H : ∀ (c : R), c x = 0c y = 0) (h : x (LinearPMap.mkSpanSingleton' x y H).domain) :
          (LinearPMap.mkSpanSingleton' x y H) { val := x, property := h } = y
          @[reducible]
          noncomputable def LinearPMap.mkSpanSingleton {K : Type u_5} {E : Type u_6} {F : Type u_7} [DivisionRing K] [AddCommGroup E] [Module K E] [AddCommGroup F] [Module K F] (x : E) (y : F) (hx : x 0) :

          The unique LinearPMap on span R {x} that sends a non-zero vector x to y. This version works for modules over division rings.

          Equations
          Instances For
            theorem LinearPMap.mkSpanSingleton_apply (K : Type u_5) {E : Type u_6} {F : Type u_7} [DivisionRing K] [AddCommGroup E] [Module K E] [AddCommGroup F] [Module K F] {x : E} (hx : x 0) (y : F) :
            (LinearPMap.mkSpanSingleton x y hx) { val := x, property := } = y
            def LinearPMap.fst {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) :
            E × F →ₗ.[R] E

            Projection to the first coordinate as a LinearPMap

            Equations
            Instances For
              @[simp]
              theorem LinearPMap.fst_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) (x : (Submodule.prod p p')) :
              (LinearPMap.fst p p') x = (x).1
              def LinearPMap.snd {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) :
              E × F →ₗ.[R] F

              Projection to the second coordinate as a LinearPMap

              Equations
              Instances For
                @[simp]
                theorem LinearPMap.snd_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) (x : (Submodule.prod p p')) :
                (LinearPMap.snd p p') x = (x).2
                instance LinearPMap.le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                LE (E →ₗ.[R] F)
                Equations
                • LinearPMap.le = { le := fun (f g : E →ₗ.[R] F) => f.domain g.domain ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄, x = yf x = g y }
                theorem LinearPMap.apply_comp_inclusion {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {T : E →ₗ.[R] F} {S : E →ₗ.[R] F} (h : T S) (x : T.domain) :
                T x = S ((Submodule.inclusion ) x)
                theorem LinearPMap.exists_of_le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {T : E →ₗ.[R] F} {S : E →ₗ.[R] F} (h : T S) (x : T.domain) :
                ∃ (y : S.domain), x = y T x = S y
                theorem LinearPMap.eq_of_le_of_domain_eq {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (hle : f g) (heq : f.domain = g.domain) :
                f = g
                def LinearPMap.eqLocus {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) :

                Given two partial linear maps f, g, the set of points x such that both f and g are defined at x and f x = g x form a submodule.

                Equations
                • One or more equations did not get rendered due to their size.
                Instances For
                  instance LinearPMap.inf {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  Inf (E →ₗ.[R] F)
                  Equations
                  instance LinearPMap.bot {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  Bot (E →ₗ.[R] F)
                  Equations
                  • LinearPMap.bot = { bot := { domain := , toFun := 0 } }
                  instance LinearPMap.inhabited {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  Equations
                  • LinearPMap.inhabited = { default := }
                  instance LinearPMap.semilatticeInf {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  Equations
                  instance LinearPMap.orderBot {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  Equations
                  theorem LinearPMap.le_of_eqLocus_ge {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (H : f.domain LinearPMap.eqLocus f g) :
                  f g
                  theorem LinearPMap.domain_mono {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                  StrictMono LinearPMap.domain
                  noncomputable def LinearPMap.sup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (h : ∀ (x : f.domain) (y : g.domain), x = yf x = g y) :

                  Given two partial linear maps that agree on the intersection of their domains, f.sup g h is the unique partial linear map on f.domain ⊔ g.domain that agrees with f and g.

                  Equations
                  Instances For
                    @[simp]
                    theorem LinearPMap.domain_sup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (h : ∀ (x : f.domain) (y : g.domain), x = yf x = g y) :
                    (LinearPMap.sup f g h).domain = f.domain g.domain
                    theorem LinearPMap.sup_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (H : ∀ (x : f.domain) (y : g.domain), x = yf x = g y) (x : f.domain) (y : g.domain) (z : (f.domain g.domain)) (hz : x + y = z) :
                    (LinearPMap.sup f g H) z = f x + g y
                    theorem LinearPMap.left_le_sup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (h : ∀ (x : f.domain) (y : g.domain), x = yf x = g y) :
                    theorem LinearPMap.right_le_sup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (h : ∀ (x : f.domain) (y : g.domain), x = yf x = g y) :
                    theorem LinearPMap.sup_le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} {h : E →ₗ.[R] F} (H : ∀ (x : f.domain) (y : g.domain), x = yf x = g y) (fh : f h) (gh : g h) :
                    theorem LinearPMap.sup_h_of_disjoint {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (h : Disjoint f.domain g.domain) (x : f.domain) (y : g.domain) (hxy : x = y) :
                    f x = g y

                    Hypothesis for LinearPMap.sup holds, if f.domain is disjoint with g.domain.

                    Algebraic operations #

                    instance LinearPMap.instZero {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Equations
                    • LinearPMap.instZero = { zero := { domain := , toFun := 0 } }
                    @[simp]
                    theorem LinearPMap.zero_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    0.domain =
                    @[simp]
                    theorem LinearPMap.zero_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (x : ) :
                    0 x = 0
                    instance LinearPMap.instSMul {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] :
                    SMul M (E →ₗ.[R] F)
                    Equations
                    • LinearPMap.instSMul = { smul := fun (a : M) (f : E →ₗ.[R] F) => { domain := f.domain, toFun := a f.toFun } }
                    @[simp]
                    theorem LinearPMap.smul_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] (a : M) (f : E →ₗ.[R] F) :
                    (a f).domain = f.domain
                    theorem LinearPMap.smul_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] (a : M) (f : E →ₗ.[R] F) (x : (a f).domain) :
                    (a f) x = a f x
                    @[simp]
                    theorem LinearPMap.coe_smul {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] (a : M) (f : E →ₗ.[R] F) :
                    (a f) = a f
                    instance LinearPMap.instSMulCommClass {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} {N : Type u_6} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] [Monoid N] [DistribMulAction N F] [SMulCommClass R N F] [SMulCommClass M N F] :
                    Equations
                    • =
                    instance LinearPMap.instIsScalarTower {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} {N : Type u_6} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] [Monoid N] [DistribMulAction N F] [SMulCommClass R N F] [SMul M N] [IsScalarTower M N F] :
                    Equations
                    • =
                    instance LinearPMap.instMulAction {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] :
                    Equations
                    instance LinearPMap.instNeg {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Neg (E →ₗ.[R] F)
                    Equations
                    • LinearPMap.instNeg = { neg := fun (f : E →ₗ.[R] F) => { domain := f.domain, toFun := -f.toFun } }
                    @[simp]
                    theorem LinearPMap.neg_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
                    (-f).domain = f.domain
                    @[simp]
                    theorem LinearPMap.neg_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : (-f).domain) :
                    (-f) x = -f x
                    instance LinearPMap.instInvolutiveNeg {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Equations
                    instance LinearPMap.instAdd {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Add (E →ₗ.[R] F)
                    Equations
                    • One or more equations did not get rendered due to their size.
                    theorem LinearPMap.add_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) :
                    (f + g).domain = f.domain g.domain
                    theorem LinearPMap.add_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (x : (f.domain g.domain)) :
                    (f + g) x = f { val := x, property := } + g { val := x, property := }
                    instance LinearPMap.instAddSemigroup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Equations
                    instance LinearPMap.instAddZeroClass {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Equations
                    instance LinearPMap.instAddMonoid {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Equations
                    instance LinearPMap.instAddCommMonoid {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Equations
                    instance LinearPMap.instVAdd {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    VAdd (E →ₗ[R] F) (E →ₗ.[R] F)
                    Equations
                    @[simp]
                    theorem LinearPMap.vadd_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (g : E →ₗ.[R] F) :
                    (f +ᵥ g).domain = g.domain
                    theorem LinearPMap.vadd_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (g : E →ₗ.[R] F) (x : (f +ᵥ g).domain) :
                    (f +ᵥ g) x = f x + g x
                    @[simp]
                    theorem LinearPMap.coe_vadd {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (g : E →ₗ.[R] F) :
                    (f +ᵥ g) = (f ∘ₗ Submodule.subtype g.domain) + g
                    instance LinearPMap.instAddAction {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Equations
                    instance LinearPMap.instSub {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Sub (E →ₗ.[R] F)
                    Equations
                    • One or more equations did not get rendered due to their size.
                    theorem LinearPMap.sub_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) :
                    (f - g).domain = f.domain g.domain
                    theorem LinearPMap.sub_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (g : E →ₗ.[R] F) (x : (f.domain g.domain)) :
                    (f - g) x = f { val := x, property := } - g { val := x, property := }
                    instance LinearPMap.instSubtractionCommMonoid {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] :
                    Equations
                    noncomputable def LinearPMap.supSpanSingleton {E : Type u_2} [AddCommGroup E] {F : Type u_3} [AddCommGroup F] {K : Type u_5} [DivisionRing K] [Module K E] [Module K F] (f : E →ₗ.[K] F) (x : E) (y : F) (hx : xf.domain) :

                    Extend a LinearPMap to f.domain ⊔ K ∙ x.

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                      @[simp]
                      theorem LinearPMap.domain_supSpanSingleton {E : Type u_2} [AddCommGroup E] {F : Type u_3} [AddCommGroup F] {K : Type u_5} [DivisionRing K] [Module K E] [Module K F] (f : E →ₗ.[K] F) (x : E) (y : F) (hx : xf.domain) :
                      (LinearPMap.supSpanSingleton f x y hx).domain = f.domain Submodule.span K {x}
                      @[simp]
                      theorem LinearPMap.supSpanSingleton_apply_mk {E : Type u_2} [AddCommGroup E] {F : Type u_3} [AddCommGroup F] {K : Type u_5} [DivisionRing K] [Module K E] [Module K F] (f : E →ₗ.[K] F) (x : E) (y : F) (hx : xf.domain) (x' : E) (hx' : x' f.domain) (c : K) :
                      (LinearPMap.supSpanSingleton f x y hx) { val := x' + c x, property := } = f { val := x', property := hx' } + c y
                      noncomputable def LinearPMap.sSup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (c : Set (E →ₗ.[R] F)) (hc : DirectedOn (fun (x x_1 : E →ₗ.[R] F) => x x_1) c) :
                      Equations
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                        theorem LinearPMap.le_sSup {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {c : Set (E →ₗ.[R] F)} (hc : DirectedOn (fun (x x_1 : E →ₗ.[R] F) => x x_1) c) {f : E →ₗ.[R] F} (hf : f c) :
                        theorem LinearPMap.sSup_le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {c : Set (E →ₗ.[R] F)} (hc : DirectedOn (fun (x x_1 : E →ₗ.[R] F) => x x_1) c) {g : E →ₗ.[R] F} (hg : fc, f g) :
                        theorem LinearPMap.sSup_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {c : Set (E →ₗ.[R] F)} (hc : DirectedOn (fun (x x_1 : E →ₗ.[R] F) => x x_1) c) {l : E →ₗ.[R] F} (hl : l c) (x : l.domain) :
                        (LinearPMap.sSup c hc) { val := x, property := } = l x
                        def LinearMap.toPMap {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (p : Submodule R E) :

                        Restrict a linear map to a submodule, reinterpreting the result as a LinearPMap.

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                          @[simp]
                          theorem LinearMap.toPMap_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (p : Submodule R E) (x : p) :
                          (LinearMap.toPMap f p) x = f x
                          @[simp]
                          theorem LinearMap.toPMap_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ[R] F) (p : Submodule R E) :
                          (LinearMap.toPMap f p).domain = p
                          def LinearMap.compPMap {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {G : Type u_4} [AddCommGroup G] [Module R G] (g : F →ₗ[R] G) (f : E →ₗ.[R] F) :

                          Compose a linear map with a LinearPMap

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                            @[simp]
                            theorem LinearMap.compPMap_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {G : Type u_4} [AddCommGroup G] [Module R G] (g : F →ₗ[R] G) (f : E →ₗ.[R] F) (x : (LinearMap.compPMap g f).domain) :
                            (LinearMap.compPMap g f) x = g (f x)
                            def LinearPMap.codRestrict {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (p : Submodule R F) (H : ∀ (x : f.domain), f x p) :
                            E →ₗ.[R] p

                            Restrict codomain of a LinearPMap

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                              def LinearPMap.comp {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {G : Type u_4} [AddCommGroup G] [Module R G] (g : F →ₗ.[R] G) (f : E →ₗ.[R] F) (H : ∀ (x : f.domain), f x g.domain) :

                              Compose two LinearPMaps

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                                def LinearPMap.coprod {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {G : Type u_4} [AddCommGroup G] [Module R G] (f : E →ₗ.[R] G) (g : F →ₗ.[R] G) :
                                E × F →ₗ.[R] G

                                f.coprod g is the partially defined linear map defined on f.domain × g.domain, and sending p to f p.1 + g p.2.

                                Equations
                                • One or more equations did not get rendered due to their size.
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                                  @[simp]
                                  theorem LinearPMap.coprod_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {G : Type u_4} [AddCommGroup G] [Module R G] (f : E →ₗ.[R] G) (g : F →ₗ.[R] G) (x : (LinearPMap.coprod f g).domain) :
                                  (LinearPMap.coprod f g) x = f { val := (x).1, property := } + g { val := (x).2, property := }
                                  def LinearPMap.domRestrict {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (S : Submodule R E) :

                                  Restrict a partially defined linear map to a submodule of E contained in f.domain.

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                                    @[simp]
                                    theorem LinearPMap.domRestrict_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {S : Submodule R E} :
                                    (LinearPMap.domRestrict f S).domain = S f.domain
                                    theorem LinearPMap.domRestrict_apply {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {S : Submodule R E} ⦃x : (S f.domain) ⦃y : f.domain (h : x = y) :
                                    (LinearPMap.domRestrict f S) x = f y
                                    theorem LinearPMap.domRestrict_le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {S : Submodule R E} :

                                    Graph #

                                    def LinearPMap.graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
                                    Submodule R (E × F)

                                    The graph of a LinearPMap viewed as a submodule on E × F.

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                                      theorem LinearPMap.mem_graph_iff' {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {x : E × F} :
                                      x LinearPMap.graph f ∃ (y : f.domain), (y, f y) = x
                                      @[simp]
                                      theorem LinearPMap.mem_graph_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {x : E × F} :
                                      x LinearPMap.graph f ∃ (y : f.domain), y = x.1 f y = x.2
                                      theorem LinearPMap.mem_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) (x : f.domain) :
                                      (x, f x) LinearPMap.graph f

                                      The tuple (x, f x) is contained in the graph of f.

                                      theorem LinearPMap.graph_map_fst_eq_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
                                      theorem LinearPMap.graph_map_snd_eq_range {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :
                                      theorem LinearPMap.smul_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {M : Type u_5} [Monoid M] [DistribMulAction M F] [SMulCommClass R M F] (f : E →ₗ.[R] F) (z : M) :
                                      LinearPMap.graph (z f) = Submodule.map (LinearMap.prodMap LinearMap.id (z LinearMap.id)) (LinearPMap.graph f)

                                      The graph of z • f as a pushforward.

                                      theorem LinearPMap.neg_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :

                                      The graph of -f as a pushforward.

                                      theorem LinearPMap.mem_graph_snd_inj {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {x : E} {y : E} {x' : F} {y' : F} (hx : (x, x') LinearPMap.graph f) (hy : (y, y') LinearPMap.graph f) (hxy : x = y) :
                                      x' = y'
                                      theorem LinearPMap.mem_graph_snd_inj' {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {x : E × F} {y : E × F} (hx : x LinearPMap.graph f) (hy : y LinearPMap.graph f) (hxy : x.1 = y.1) :
                                      x.2 = y.2
                                      theorem LinearPMap.graph_fst_eq_zero_snd {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) {x : E} {x' : F} (h : (x, x') LinearPMap.graph f) (hx : x = 0) :
                                      x' = 0

                                      The property that f 0 = 0 in terms of the graph.

                                      theorem LinearPMap.mem_domain_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {x : E} :
                                      x f.domain ∃ (y : F), (x, y) LinearPMap.graph f
                                      theorem LinearPMap.mem_domain_of_mem_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {x : E} {y : F} (h : (x, y) LinearPMap.graph f) :
                                      x f.domain
                                      theorem LinearPMap.image_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {x : E} {y : F} (hx : x f.domain) :
                                      y = f { val := x, property := hx } (x, y) LinearPMap.graph f
                                      theorem LinearPMap.mem_range_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {y : F} :
                                      y Set.range f ∃ (x : E), (x, y) LinearPMap.graph f
                                      theorem LinearPMap.mem_domain_iff_of_eq_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (h : LinearPMap.graph f = LinearPMap.graph g) {x : E} :
                                      x f.domain x g.domain
                                      theorem LinearPMap.le_of_le_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (h : LinearPMap.graph f LinearPMap.graph g) :
                                      f g
                                      theorem LinearPMap.le_graph_of_le {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (h : f g) :
                                      theorem LinearPMap.le_graph_iff {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} :
                                      theorem LinearPMap.eq_of_eq_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} {g : E →ₗ.[R] F} (h : LinearPMap.graph f = LinearPMap.graph g) :
                                      f = g
                                      theorem Submodule.existsUnique_from_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {g : Submodule R (E × F)} (hg : ∀ {x : E × F}, x gx.1 = 0x.2 = 0) {a : E} (ha : a Submodule.map (LinearMap.fst R E F) g) :
                                      ∃! (b : F), (a, b) g
                                      noncomputable def Submodule.valFromGraph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {g : Submodule R (E × F)} (hg : xg, x.1 = 0x.2 = 0) {a : E} (ha : a Submodule.map (LinearMap.fst R E F) g) :
                                      F

                                      Auxiliary definition to unfold the existential quantifier.

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                                        theorem Submodule.valFromGraph_mem {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {g : Submodule R (E × F)} (hg : xg, x.1 = 0x.2 = 0) {a : E} (ha : a Submodule.map (LinearMap.fst R E F) g) :
                                        noncomputable def Submodule.toLinearPMapAux {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (g : Submodule R (E × F)) (hg : xg, x.1 = 0x.2 = 0) :

                                        Define a LinearMap from its graph.

                                        Helper definition for LinearPMap.

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                                          noncomputable def Submodule.toLinearPMap {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (g : Submodule R (E × F)) :

                                          Define a LinearPMap from its graph.

                                          In the case that the submodule is not a graph of a LinearPMap then the underlying linear map is just the zero map.

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                                            theorem Submodule.toLinearPMap_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (g : Submodule R (E × F)) :
                                            theorem Submodule.toLinearPMap_apply_aux {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {g : Submodule R (E × F)} (hg : xg, x.1 = 0x.2 = 0) (x : (Submodule.map (LinearMap.fst R E F) g)) :
                                            theorem Submodule.mem_graph_toLinearPMap {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {g : Submodule R (E × F)} (hg : xg, x.1 = 0x.2 = 0) (x : (Submodule.map (LinearMap.fst R E F) g)) :
                                            (x, (Submodule.toLinearPMap g) x) g
                                            @[simp]
                                            theorem Submodule.toLinearPMap_graph_eq {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (g : Submodule R (E × F)) (hg : xg, x.1 = 0x.2 = 0) :
                                            theorem Submodule.toLinearPMap_range {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (g : Submodule R (E × F)) (hg : xg, x.1 = 0x.2 = 0) :
                                            noncomputable def LinearPMap.inverse {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] (f : E →ₗ.[R] F) :

                                            The inverse of a LinearPMap.

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                                              theorem LinearPMap.inverse_domain {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} :
                                              theorem LinearPMap.mem_inverse_graph_snd_eq_zero {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ) (x : F × E) (hv : x Submodule.map (LinearEquiv.prodComm R E F) (LinearPMap.graph f)) (hv' : x.1 = 0) :
                                              x.2 = 0

                                              The graph of the inverse generates a LinearPMap.

                                              theorem LinearPMap.inverse_range {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ) :
                                              theorem LinearPMap.mem_inverse_graph {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ) (x : f.domain) :
                                              theorem LinearPMap.inverse_apply_eq {R : Type u_1} [Ring R] {E : Type u_2} [AddCommGroup E] [Module R E] {F : Type u_3} [AddCommGroup F] [Module R F] {f : E →ₗ.[R] F} (hf : LinearMap.ker f.toFun = ) {y : (LinearPMap.inverse f).domain} {x : f.domain} (hxy : f x = y) :
                                              (LinearPMap.inverse f) y = x