Documentation

Mathlib.Algebra.Order.Ring.InjSurj

Pulling back ordered rings along injective maps. #

@[reducible]
def Function.Injective.orderedSemiring {α : Type u} {β : Type u_1} [OrderedSemiring α] [Zero β] [One β] [Add β] [Mul β] [Pow β ] [SMul β] [NatCast β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) :

Pullback an OrderedSemiring under an injective map.

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    @[reducible]
    def Function.Injective.orderedCommSemiring {α : Type u} {β : Type u_1} [OrderedCommSemiring α] [Zero β] [One β] [Add β] [Mul β] [Pow β ] [SMul β] [NatCast β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) :

    Pullback an OrderedCommSemiring under an injective map.

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    • One or more equations did not get rendered due to their size.
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      @[reducible]
      def Function.Injective.orderedRing {α : Type u} {β : Type u_1} [OrderedRing α] [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [SMul β] [SMul β] [Pow β ] [NatCast β] [IntCast β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (zsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) (int_cast : ∀ (n : ), f n = n) :

      Pullback an OrderedRing under an injective map.

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      • One or more equations did not get rendered due to their size.
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        @[reducible]
        def Function.Injective.orderedCommRing {α : Type u} {β : Type u_1} [OrderedCommRing α] [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [Pow β ] [SMul β] [SMul β] [NatCast β] [IntCast β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (zsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) (int_cast : ∀ (n : ), f n = n) :

        Pullback an OrderedCommRing under an injective map.

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        • One or more equations did not get rendered due to their size.
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          @[reducible]
          def Function.Injective.strictOrderedSemiring {α : Type u} {β : Type u_1} [StrictOrderedSemiring α] [Zero β] [One β] [Add β] [Mul β] [Pow β ] [SMul β] [NatCast β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) :

          Pullback a StrictOrderedSemiring under an injective map.

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            @[reducible]
            def Function.Injective.strictOrderedCommSemiring {α : Type u} {β : Type u_1} [StrictOrderedCommSemiring α] [Zero β] [One β] [Add β] [Mul β] [Pow β ] [SMul β] [NatCast β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) :

            Pullback a strictOrderedCommSemiring under an injective map.

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            • One or more equations did not get rendered due to their size.
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              @[reducible]
              def Function.Injective.strictOrderedRing {α : Type u} {β : Type u_1} [StrictOrderedRing α] [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [SMul β] [SMul β] [Pow β ] [NatCast β] [IntCast β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (zsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) (int_cast : ∀ (n : ), f n = n) :

              Pullback a StrictOrderedRing under an injective map.

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              • One or more equations did not get rendered due to their size.
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                @[reducible]
                def Function.Injective.strictOrderedCommRing {α : Type u} {β : Type u_1} [StrictOrderedCommRing α] [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [Pow β ] [SMul β] [SMul β] [NatCast β] [IntCast β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (zsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) (int_cast : ∀ (n : ), f n = n) :

                Pullback a StrictOrderedCommRing under an injective map.

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                • One or more equations did not get rendered due to their size.
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                  @[reducible]
                  def Function.Injective.linearOrderedSemiring {α : Type u} {β : Type u_1} [LinearOrderedSemiring α] [Zero β] [One β] [Add β] [Mul β] [Pow β ] [SMul β] [NatCast β] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) (sup : ∀ (x y : β), f (x y) = max (f x) (f y)) (inf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

                  Pullback a LinearOrderedSemiring under an injective map.

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                  • One or more equations did not get rendered due to their size.
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                    @[reducible]
                    def Function.Injective.linearOrderedCommSemiring {α : Type u} {β : Type u_1} [LinearOrderedCommSemiring α] [Zero β] [One β] [Add β] [Mul β] [Pow β ] [SMul β] [NatCast β] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) (hsup : ∀ (x y : β), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

                    Pullback a LinearOrderedSemiring under an injective map.

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                    • One or more equations did not get rendered due to their size.
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                      @[reducible]
                      def Function.Injective.linearOrderedRing {α : Type u} {β : Type u_1} [LinearOrderedRing α] [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [SMul β] [SMul β] [Pow β ] [NatCast β] [IntCast β] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (zsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) (int_cast : ∀ (n : ), f n = n) (hsup : ∀ (x y : β), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

                      Pullback a LinearOrderedRing under an injective map.

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                      • One or more equations did not get rendered due to their size.
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                        @[reducible]
                        def Function.Injective.linearOrderedCommRing {α : Type u} {β : Type u_1} [LinearOrderedCommRing α] [Zero β] [One β] [Add β] [Mul β] [Neg β] [Sub β] [Pow β ] [SMul β] [SMul β] [NatCast β] [IntCast β] [Sup β] [Inf β] (f : βα) (hf : Function.Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : ∀ (x y : β), f (x + y) = f x + f y) (mul : ∀ (x y : β), f (x * y) = f x * f y) (neg : ∀ (x : β), f (-x) = -f x) (sub : ∀ (x y : β), f (x - y) = f x - f y) (nsmul : ∀ (x : β) (n : ), f (n x) = n f x) (zsmul : ∀ (x : β) (n : ), f (n x) = n f x) (npow : ∀ (x : β) (n : ), f (x ^ n) = f x ^ n) (nat_cast : ∀ (n : ), f n = n) (int_cast : ∀ (n : ), f n = n) (sup : ∀ (x y : β), f (x y) = max (f x) (f y)) (inf : ∀ (x y : β), f (x y) = min (f x) (f y)) :

                        Pullback a LinearOrderedCommRing under an injective map.

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                        • One or more equations did not get rendered due to their size.
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