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chore(linear_algebra/basic): add linear_equiv.conj_apply_apply (#17364
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While this lemma follows by `simp` via `conj_apply`, it is very annoying to clean up in a chain of rewrites due to having to commute all the coercions.
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eric-wieser committed Nov 6, 2022
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3 changes: 3 additions & 0 deletions src/linear_algebra/basic.lean
Expand Up @@ -1868,6 +1868,9 @@ def conj (e : M ≃ₗ[R] M₂) : (module.End R M) ≃ₗ[R] (module.End R M₂)
lemma conj_apply (e : M ≃ₗ[R] M₂) (f : module.End R M) :
e.conj f = ((↑e : M →ₗ[R] M₂).comp f).comp (e.symm : M₂ →ₗ[R] M) := rfl

lemma conj_apply_apply (e : M ≃ₗ[R] M₂) (f : module.End R M) (x : M₂) :
e.conj f x = e (f (e.symm x)) := rfl

lemma symm_conj_apply (e : M ≃ₗ[R] M₂) (f : module.End R M₂) :
e.symm.conj f = ((↑e.symm : M₂ →ₗ[R] M).comp f).comp (e : M →ₗ[R] M₂) := rfl

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