File: Mathlib/Analysis/NormedSpace/OperatorNorm/Prod.lean

full_name: ContinuousLinearMap.norm_fst_le
code: /-- The operator norm of the first projection `E × F → E` is at most 1. (It is 0 if `E` is zero, so
the inequality cannot be improved without further assumptions.) -/
lemma norm_fst_le : ‖fst 𝕜 E F‖ ≤ 1 
goal: ‖ContinuousLinearMap.fst 𝕜 E F‖ ≤ 1

full_name: ContinuousLinearMap.norm_snd_le
code: /-- The operator norm of the second projection `E × F → F` is at most 1. (It is 0 if `F` is zero, so
the inequality cannot be improved without further assumptions.) -/
lemma norm_snd_le : ‖snd 𝕜 E F‖ ≤ 1 
goal: ‖ContinuousLinearMap.snd 𝕜 E F‖ ≤ 1

full_name: ContinuousLinearMap.opNorm_prod
code: @[simp]
theorem opNorm_prod (f : E →L[𝕜] F) (g : E →L[𝕜] G) : ‖f.prod g‖ = ‖(f, g)‖ 
goal: ‖f.prod g‖ = ‖(f, g)‖

full_name: ContinuousLinearMap.opNNNorm_prod
code: @[simp]
theorem opNNNorm_prod (f : E →L[𝕜] F) (g : E →L[𝕜] G) : ‖f.prod g‖₊ = ‖(f, g)‖₊ 
goal: ‖f.prod g‖₊ = ‖(f, g)‖₊

full_name: ContinuousLinearMap.prodₗᵢ
code: /-- `ContinuousLinearMap.prod` as a `LinearIsometryEquiv`. -/
def prodₗᵢ (R : Type*) [Semiring R] [Module R F] [Module R G] [ContinuousConstSMul R F]
    [ContinuousConstSMul R G] [SMulCommClass 𝕜 R F] [SMulCommClass 𝕜 R G] :
    (E →L[𝕜] F) × (E →L[𝕜] G) ≃ₗᵢ[R] E →L[𝕜] F × G 
goal: (E →L[𝕜] F) × (E →L[𝕜] G) ≃ₗᵢ[R] E →L[𝕜] F × G

full_name: ContinuousLinearMap.prodMapL
code: /-- `ContinuousLinearMap.prodMap` as a continuous linear map. -/
def prodMapL : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄) →L[𝕜] M₁ × M₃ →L[𝕜] M₂ × M₄ 
goal: (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄) →L[𝕜] M₁ × M₃ →L[𝕜] M₂ × M₄

full_name: ContinuousLinearMap.prodMapL_apply
code: @[simp]
theorem prodMapL_apply (p : (M₁ →L[𝕜] M₂) × (M₃ →L[𝕜] M₄)) :
    ContinuousLinearMap.prodMapL 𝕜 M₁ M₂ M₃ M₄ p = p.1.prodMap p.2 
goal: (ContinuousLinearMap.prodMapL 𝕜 M₁ M₂ M₃ M₄) p = p.1.prodMap p.2

full_name: Continuous.prod_mapL
code: theorem _root_.Continuous.prod_mapL {f : X → M₁ →L[𝕜] M₂} {g : X → M₃ →L[𝕜] M₄} (hf : Continuous f)
    (hg : Continuous g) : Continuous fun x => (f x).prodMap (g x) 
goal: Continuous fun x => (f x).prodMap (g x)

full_name: Continuous.prod_map_equivL
code: theorem _root_.Continuous.prod_map_equivL {f : X → M₁ ≃L[𝕜] M₂} {g : X → M₃ ≃L[𝕜] M₄}
    (hf : Continuous fun x => (f x : M₁ →L[𝕜] M₂)) (hg : Continuous fun x => (g x : M₃ →L[𝕜] M₄)) :
    Continuous fun x => ((f x).prod (g x) : M₁ × M₃ →L[𝕜] M₂ × M₄) 
goal: Continuous fun x => ↑((f x).prod (g x))

full_name: ContinuousOn.prod_mapL
code: theorem _root_.ContinuousOn.prod_mapL {f : X → M₁ →L[𝕜] M₂} {g : X → M₃ →L[𝕜] M₄} {s : Set X}
    (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
    ContinuousOn (fun x => (f x).prodMap (g x)) s 
goal: ContinuousOn (fun x => (f x).prodMap (g x)) s

full_name: ContinuousOn.prod_map_equivL
code: theorem _root_.ContinuousOn.prod_map_equivL {f : X → M₁ ≃L[𝕜] M₂} {g : X → M₃ ≃L[𝕜] M₄} {s : Set X}
    (hf : ContinuousOn (fun x => (f x : M₁ →L[𝕜] M₂)) s)
    (hg : ContinuousOn (fun x => (g x : M₃ →L[𝕜] M₄)) s) :
    ContinuousOn (fun x => ((f x).prod (g x) : M₁ × M₃ →L[𝕜] M₂ × M₄)) s 
goal: ContinuousOn (fun x => ↑((f x).prod (g x))) s

full_name: ContinuousLinearMap.norm_fst
code: /-- The operator norm of the first projection `E × F → E` is exactly 1 if `E` is nontrivial. -/
@[simp] lemma norm_fst [NormedAddCommGroup E] [NormedSpace 𝕜 E]
    [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] [Nontrivial E] :
    ‖fst 𝕜 E F‖ = 1 
goal: ‖ContinuousLinearMap.fst 𝕜 E F‖ = 1

full_name: ContinuousLinearMap.norm_snd
code: /-- The operator norm of the second projection `E × F → F` is exactly 1 if `F` is nontrivial. -/
@[simp] lemma norm_snd [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
    [NormedAddCommGroup F] [NormedSpace 𝕜 F] [Nontrivial F]  :
    ‖snd 𝕜 E F‖ = 1 
goal: ‖ContinuousLinearMap.snd 𝕜 E F‖ = 1