Ring.closure.isSubring
theorem Ring.closure.isSubring : ∀ {R : Type u} [inst : Ring R] {s : Set R}, IsSubring (Ring.closure s)
The theorem `Ring.closure.isSubring` states that for any type `R` with a ring structure, and any set `s` of elements of `R`, the smallest subring of `R` that contains `s` (denoted by `Ring.closure s`) is indeed a subring of `R`. In mathematical terms, given a ring $(R, +, \cdot)$ and a subset $s$ of $R$, the closure of $s$ under the ring operations (addition and multiplication), denoted by `Ring.closure s`, forms a subring of $R$.
More concisely: Let $(R, +, \cdot)$ be a ring and $s$ be a subset of $R$. The set `Ring.closure s` is a subring of $R$.
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Ring.closure_subset
theorem Ring.closure_subset : ∀ {R : Type u} [inst : Ring R] {s t : Set R}, IsSubring t → s ⊆ t → Ring.closure s ⊆ t
This theorem states that for any type `R` that forms a ring, given two sets `s` and `t` of elements from `R`, if `t` forms a subring and all elements of `s` are also elements of `t`, then all elements of the smallest subring (as defined by `Ring.closure`) containing `s` are also elements of `t`. In other words, the closure of `s` under the ring operations is a subset of `t`.
In mathematical terms, if `t` is a subring of a ring `R` and `s` is a subset of `t`, then the closure of `s` (i.e., the smallest subring containing `s`) is also a subset of `t`.
More concisely: If `t` is a subring of a ring `R` and `s` is a subset of `t`, then the closure of `s` in `R` is contained in `t`.
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