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feat(algebra/*): morphisms from closures are equal if they agree on generators #18836
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eric-wieser
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eric-wieser
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Apr 19, 2023
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| lemma ext_adjoin {s : set A} [star_alg_hom_class F R (adjoin R s) B] {f g : F} | ||
| (h : ∀ x : adjoin R s, (x : A) ∈ s → f x = g x) : f = g := | ||
| /-- Two star algebra morphisms from `star_subalgebra.adjoin` are equal if they agree on the | ||
| generators -/ | ||
| lemma _root_.star_alg_hom_class.ext_adjoin | ||
| {s : set A} [star_alg_hom_class F R (adjoin R s) B] ⦃f g : F⦄ | ||
| (h : f ∘ set.inclusion (subset_adjoin _ _) = g ∘ set.inclusion (subset_adjoin _ _)) : f = g := |
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@j-loreaux, do you have thoughts on which phrasing is more useful of:
f ∘ set.inclusion (subset_adjoin _ _) = g ∘ set.inclusion (subset_adjoin _ _)∀ x : adjoin R s, (x : A) ∈ s → f x = g x∀ (x : A) (hx : x ∈ s), f ⟨x, subset_adjoin _ _ hx⟩ = g ⟨x, subset_adjoin _ _ hx⟩
In theory the first one lets you chain further ext lemmas, but in practice I don't think any exist.
eric-wieser
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Apr 19, 2023
| generators. | ||
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| See note [partially-applied ext lemmas]. -/ | ||
| @[ext] lemma ext_adjoin {s : set A} ⦃f g : adjoin R s →⋆ₐ[R] B⦄ |
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Unfortunately the existing lemma couldn't be tagged ext
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This adds this statement for:
subsemigroup,add_subsemigroupsubmonoid,add_submonoidsubmodulesubalgebrastar_subalgebraI don't add it for
subsemiringorsubringas these are missing theinduction'lemma used to prove it.