refactor(LinearAlgebra/BilinearMap): Congruences and bilinear maps - #8255
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mans0954 wants to merge 13 commits into
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refactor(LinearAlgebra/BilinearMap): Congruences and bilinear maps#8255mans0954 wants to merge 13 commits into
mans0954 wants to merge 13 commits into
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j-loreaux
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Dec 4, 2023
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| /-- Apply linear equivalences on the arguments of a bilinear map. -/ | ||
| def congrl₁₂ (e₁ : Mₗ ≃ₗ[R] Nₗ) (e₂ : Pₗ ≃ₗ[R] Qₗ) : | ||
| (Mₗ →ₗ[R] Pₗ →ₗ[R] M) ≃ₗ[R] (Nₗ →ₗ[R] Qₗ →ₗ[R] M) where |
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I'm not convinced we need this at all, as it's just two applications of LinearEquiv.arrowCongr. And even if we did want it, it should take a third equivalence as an argument and be called LinearEquiv.arrowCongr₂
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| /-- Apply linear equivalences on the arguments of a bilinear map. -/ | |
| def congrl₁₂ (e₁ : Mₗ ≃ₗ[R] Nₗ) (e₂ : Pₗ ≃ₗ[R] Qₗ) : | |
| (Mₗ →ₗ[R] Pₗ →ₗ[R] M) ≃ₗ[R] (Nₗ →ₗ[R] Qₗ →ₗ[R] M) where | |
| /-- Apply linear equivalences on the arguments of a bilinear map. -/ | |
| def congrl₁₂ (e₁ : Mₗ ≃ₗ[R] Nₗ) (e₂ : Pₗ ≃ₗ[R] Qₗ) : | |
| (Mₗ →ₗ[R] Pₗ →ₗ[R] M) ≃ₗ[R] (Nₗ →ₗ[R] Qₗ →ₗ[R] M) := | |
| e₁.arrowCongr <| e₂.arrowCongr <| .refl R M |
(Note this is definitionally equal to what you wrote.) If you want, you can update the definition of BilinForm.congr to use LinearEquiv.arrowCongr
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Not needed. |
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Currently Mathlib has
BilinForm.congrwhich defines a congruence between bilinear forms on the domain and range of an equivalence. This PR generalises this result to bilinear maps. The bilinear form version is recovered as a special case of the more general result.Required for #8256