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[Merged by Bors] - feat: add @[simp] lemma Prod.norm_mk #20411

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[Merged by Bors] - feat: add @[simp] lemma Prod.norm_mk #20411

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7b7d0fb
feat: add `@[simp] lemma Prod.norm_mk`
urkud Jan 2, 2025
1a60880
Fix
urkud Jan 2, 2025
88c3819
Fix
urkud Jan 3, 2025
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14 changes: 12 additions & 2 deletions Mathlib/Analysis/Normed/Group/Constructions.lean
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Expand Up @@ -229,6 +229,8 @@ instance Prod.toNorm : Norm (E × F) where norm x := ‖x.1‖ ⊔ ‖x.2‖

lemma Prod.norm_def (x : E × F) : ‖x‖ = max ‖x.1‖ ‖x.2‖ := rfl

@[simp] lemma Prod.norm_mk (x : E) (y : F) : ‖(x, y)‖ = max ‖x‖ ‖y‖ := rfl
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lemma norm_fst_le (x : E × F) : ‖x.1‖ ≤ ‖x‖ := le_max_left _ _

lemma norm_snd_le (x : E × F) : ‖x.2‖ ≤ ‖x‖ := le_max_right _ _
Expand All @@ -246,8 +248,16 @@ instance Prod.seminormedGroup : SeminormedGroup (E × F) where
dist_eq x y := by
simp only [Prod.norm_def, Prod.dist_eq, dist_eq_norm_div, Prod.fst_div, Prod.snd_div]

@[to_additive Prod.nnnorm_def']
lemma Prod.nnorm_def (x : E × F) : ‖x‖₊ = max ‖x.1‖₊ ‖x.2‖₊ := rfl
/-- Multiplicative version of `Prod.nnnorm_def`.
Earlier, this names was used for the additive version. -/
@[to_additive Prod.nnnorm_def]
lemma Prod.nnnorm_def' (x : E × F) : ‖x‖₊ = max ‖x.1‖₊ ‖x.2‖₊ := rfl

@[deprecated (since := "2025-01-02")] alias Prod.nnorm_def := Prod.nnnorm_def'

/-- Multiplicative version of `Prod.nnnorm_mk`. -/
@[to_additive (attr := simp) Prod.nnnorm_mk]
lemma Prod.nnnorm_mk' (x : E) (y : F) : ‖(x, y)‖₊ = max ‖x‖₊ ‖y‖₊ := rfl

end SeminormedGroup

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2 changes: 1 addition & 1 deletion Mathlib/Analysis/Normed/Lp/ProdLp.lean
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Expand Up @@ -616,7 +616,7 @@ theorem prod_nnnorm_eq_sup (f : WithLp ∞ (α × β)) : ‖f‖₊ = ‖f.fst
norm_cast

@[simp] theorem prod_nnnorm_equiv (f : WithLp ∞ (α × β)) : ‖WithLp.equiv ⊤ _ f‖₊ = ‖f‖₊ := by
rw [prod_nnnorm_eq_sup, Prod.nnnorm_def', equiv_fst, equiv_snd]
rw [prod_nnnorm_eq_sup, Prod.nnnorm_def, equiv_fst, equiv_snd]

@[simp] theorem prod_nnnorm_equiv_symm (f : α × β) : ‖(WithLp.equiv ⊤ _).symm f‖₊ = ‖f‖₊ :=
(prod_nnnorm_equiv _).symm
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2 changes: 1 addition & 1 deletion Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
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Expand Up @@ -577,7 +577,7 @@ theorem isLeast_opNNNorm (f : ContinuousMultilinearMap 𝕜 E G) :
theorem opNNNorm_prod (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G') :
‖f.prod g‖₊ = max ‖f‖₊ ‖g‖₊ :=
eq_of_forall_ge_iff fun _ ↦ by
simp only [opNNNorm_le_iff, prod_apply, Prod.nnnorm_def', max_le_iff, forall_and]
simp only [opNNNorm_le_iff, prod_apply, Prod.nnnorm_def, max_le_iff, forall_and]

@[deprecated (since := "2024-02-02")] alias op_nnnorm_prod := opNNNorm_prod

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2 changes: 1 addition & 1 deletion Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
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Expand Up @@ -388,7 +388,7 @@ theorem nnnorm_eq_sup_normAtPlace (x : mixedSpace K) :
(univ.image (fun w : {w : InfinitePlace K // IsComplex w} ↦ w.1)) := by
ext; simp [isReal_or_isComplex]
rw [this, sup_union, univ.sup_image, univ.sup_image,
Prod.nnnorm_def', Pi.nnnorm_def, Pi.nnnorm_def]
Prod.nnnorm_def, Pi.nnnorm_def, Pi.nnnorm_def]
congr
· ext w
simp [normAtPlace_apply_isReal w.prop]
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