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[Merged by Bors] - chore(RingTheory/AdicCompletion): update the definition of AdicCompletion.map #35595

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BryceT233 wants to merge 1 commit into from
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[Merged by Bors] - chore(RingTheory/AdicCompletion): update the definition of AdicCompletion.map #35595

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38 changes: 20 additions & 18 deletions Mathlib/RingTheory/AdicCompletion/Functoriality.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -133,24 +133,6 @@ private theorem adicCompletionAux_val_apply (f : M →ₗ[R] N) {n : ℕ} (x : A
(adicCompletionAux I f x).val n = f.reduceModIdeal (I ^ n) (x.val n) :=
rfl

set_option backward.isDefEq.respectTransparency false in
set_option backward.privateInPublic true in
set_option backward.privateInPublic.warn false in
/-- A linear map induces a map on adic completions. -/
def map (f : M →ₗ[R] N) :
AdicCompletion I M →ₗ[AdicCompletion I R] AdicCompletion I N where
toFun := adicCompletionAux I f
map_add' := by simp
map_smul' r x := by
ext n
simp only [adicCompletionAux_val_apply, smul_eval, smul_eq_mul, RingHom.id_apply]
rw [val_smul_eq_evalₐ_smul, val_smul_eq_evalₐ_smul, map_smul]

@[simp]
theorem map_val_apply (f : M →ₗ[R] N) {n : ℕ} (x : AdicCompletion I M) :
(map I f x).val n = f.reduceModIdeal (I ^ n) (x.val n) :=
rfl

/-- Equality of maps out of an adic completion can be checked on Cauchy sequences. -/
theorem map_ext {N} {f g : AdicCompletion I M → N}
(h : ∀ (a : AdicCauchySequence I M),
Expand All @@ -176,6 +158,26 @@ theorem map_ext'' {f g : AdicCompletion I M →ₗ[R] N}
ext x
apply induction_on I M x (fun a ↦ LinearMap.ext_iff.mp h a)

set_option backward.isDefEq.respectTransparency false in
set_option backward.privateInPublic true in
set_option backward.privateInPublic.warn false in
/-- A linear map induces a map on adic completions. -/
def map : (M →ₗ[R] N) →ₗ[R] AdicCompletion I M →ₗ[AdicCompletion I R] AdicCompletion I N where
toFun f :=
{ toFun := adicCompletionAux I f
map_add' := by simp
map_smul' r x := by
ext n
simp only [adicCompletionAux_val_apply, smul_eval, smul_eq_mul, RingHom.id_apply]
rw [val_smul_eq_evalₐ_smul, val_smul_eq_evalₐ_smul, map_smul] }
map_add' _ _ := by ext; rfl
map_smul' _ _ := by ext; rfl

@[simp]
theorem map_val_apply (f : M →ₗ[R] N) {n : ℕ} (x : AdicCompletion I M) :
(map I f x).val n = f.reduceModIdeal (I ^ n) (x.val n) :=
rfl

variable (M) in
@[simp]
theorem map_id :
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