mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E✝ : Type u_3
inst✝¹⁵ : NormedAddCommGroup E✝
inst✝¹⁴ : NormedSpace 𝕜 E✝
F✝ : Type u_4
inst✝¹³ : NormedAddCommGroup F✝
inst✝¹² : NormedSpace 𝕜' F✝
f✝¹ : E✝ →SL[σ] F✝
σ' : 𝕜' →+* 𝕜
inst✝¹¹ : RingHomInvPair σ σ'
inst✝¹⁰ : RingHomIsometric σ
inst✝⁹ : RingHomIsometric σ'
inst✝⁸ : CompleteSpace F✝
inst✝⁷ : CompleteSpace E✝
inst✝⁶ : RingHomInvPair σ' σ
f✝ : E✝ →SL[σ] F✝
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : E →L[𝕜] F
hf : Injective ⇑f
hf' : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Set.range ⇑f)
K : ℝ≥0 := f.antilipschitzConstant_of_injective_of_isClosed_range hf hf'
hfK : AntilipschitzWith (f.antilipschitzConstant_of_injective_of_isClosed_range hf hf') ⇑f :=
  antilipschitz_antiLipschitzConstant_of_injective_of_isClosed_range f hf hf'
x : E
aux : ‖x‖ ≤ ↑(f.antilipschitzConstant_of_injective_of_isClosed_range hf hf') * ‖f x‖
⊢ (↑f.rangeRestrict).ker = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
p : ℝ≥0∞
⊢ 1 - idemFst = idemSnd

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
p : ℝ≥0∞
⊢ 1 - idemSnd = idemFst

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
inst✝¹⁶ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E✝ : Type u_3
inst✝¹⁵ : NormedAddCommGroup E✝
inst✝¹⁴ : NormedSpace 𝕜 E✝
F✝ : Type u_4
inst✝¹³ : NormedAddCommGroup F✝
inst✝¹² : NormedSpace 𝕜' F✝
f✝¹ : E✝ →SL[σ] F✝
σ' : 𝕜' →+* 𝕜
inst✝¹¹ : RingHomInvPair σ σ'
inst✝¹⁰ : RingHomIsometric σ
inst✝⁹ : RingHomIsometric σ'
inst✝⁸ : CompleteSpace F✝
inst✝⁷ : CompleteSpace E✝
inst✝⁶ : RingHomInvPair σ' σ
f✝ : E✝ →SL[σ] F✝
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
f : E →L[𝕜] F
hf : Injective ⇑f
hf' : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Set.range ⇑f)
K : ℝ≥0 := f.antilipschitzConstant_of_injective_of_isClosed_range hf hf'
hfK : AntilipschitzWith (f.antilipschitzConstant_of_injective_of_isClosed_range hf hf') ⇑f :=
  antilipschitz_antiLipschitzConstant_of_injective_of_isClosed_range f hf hf'
⊢ ∀ (x : ↥(↑f).range), ‖(↑f.rangeRestrict).leftInverse x‖ ≤ ↑K * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x : WithLp ∞ (α × β)
⊢ ‖x‖ = max ‖idemFst x‖ ‖idemSnd x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜' F
f✝ : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁵ : RingHomInvPair σ σ'
inst✝⁴ : RingHomIsometric σ
inst✝³ : RingHomIsometric σ'
inst✝² : CompleteSpace F
inst✝¹ : CompleteSpace E
inst✝ : RingHomInvPair σ' σ
f : E →SL[σ] F
hinj : (↑f).ker = ⊥
hsurj : (↑f).range = ⊤
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ⇑(LinearEquiv.ofBijective ↑f ⋯)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜' F
σ' : 𝕜' →+* 𝕜
inst✝⁵ : RingHomInvPair σ σ'
inst✝⁴ : RingHomIsometric σ
inst✝³ : RingHomIsometric σ'
inst✝² : CompleteSpace F
inst✝¹ : CompleteSpace E
inst✝ : RingHomInvPair σ' σ
f : E →SL[σ] F
hinj : (↑f).ker = ⊥
hsurj : (↑f).range = ⊤
⊢ ↑(ofBijective f hinj hsurj) = f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : Fact (1 ≤ p)
inst✝² : (i : ι) → PseudoMetricSpace (α i)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
f g : PiLp p α
h : 1 ≤ p.toReal
⊢ 0 < p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝² : SeminormedAddCommGroup α
inst✝¹ : SeminormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hp : 0 < p.toReal
x : WithLp p (α × β)
⊢ ‖x‖ = (‖idemFst x‖ ^ p.toReal + ‖idemSnd x‖ ^ p.toReal) ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : Fact (1 ≤ p)
inst✝² : (i : ι) → PseudoMetricSpace (α i)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
f g : PiLp p α
h : 1 ≤ p.toReal
this : 0 < p.toReal
⊢ 0 ≤ ∑ i, dist (f.ofLp i) (g.ofLp i) ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : Fact (1 ≤ p)
inst✝² : (i : ι) → PseudoMetricSpace (α i)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
f g : PiLp p α
h : 1 ≤ p.toReal
this : 0 < p.toReal
⊢ 0 ≤ 1 / p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x : WithLp 1 (α × β)
⊢ ‖x‖ = ‖idemFst x‖ + ‖idemSnd x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: case mpr
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : E →L[𝕜] E
⊢ Bijective ⇑f → IsUnit f

Tactic:
· refine fun h ↦ ⟨toUnit <| .ofBijective f ?_ ?_, rfl⟩ <;>
    simp only [LinearMap.range_eq_top, LinearMap.ker_eq_bot, f.coe_coe, h.1, h.2]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : E →L[𝕜] E
⊢ IsUnit f ↔ Bijective ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : SeminormedAddCommGroup α
inst✝ : SeminormedAddCommGroup β
x y : α × β
⊢ dist x y = ‖toLp p (-x + y)‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
inst✝³ : Fact (1 ≤ p)
inst✝² : (i : ι) → PseudoMetricSpace (α i)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
f g : PiLp p α
⊢ edist f g = ENNReal.ofReal (dist f g)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
hp : Fact (1 ≤ p)
α : Type u_4
β : Type u_5
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
R : Type u_6
inst✝⁴ : SeminormedRing R
inst✝³ : Module R α
inst✝² : Module R β
inst✝¹ : IsBoundedSMul R α
inst✝ : IsBoundedSMul R β
⊢ IsBoundedSMul R (α × β)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
hp : Fact (1 ≤ p)
α : Type u_4
β : Type u_5
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
R : Type u_6
inst✝⁴ : SeminormedRing R
inst✝³ : Module R α
inst✝² : Module R β
inst✝¹ : NormSMulClass R α
inst✝ : NormSMulClass R β
⊢ NormSMulClass R (α × β)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α✝
inst✝⁵ : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝⁴ : SeminormedAddCommGroup α
inst✝³ : SeminormedAddCommGroup β
R : Type u_6
inst✝² : NormedField R
inst✝¹ : NormedSpace R α
inst✝ : NormedSpace R β
⊢ NormedSpace R (α × β)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : NormedAddCommGroup α
inst✝ : NormedAddCommGroup β
x y : α × β
h : dist x y = 0
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α✝ : Type u_2
β✝ : Type u_3
hp : Fact (1 ≤ p)
inst✝³ : SeminormedAddCommGroup α✝
inst✝² : SeminormedAddCommGroup β✝
α : Type u_4
β : Type u_5
inst✝¹ : NormedAddCommGroup α
inst✝ : NormedAddCommGroup β
x y : α × β
⊢ dist x y = ‖toLp p (-x + y)‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : E →L[𝕜] E
⊢ spectrum 𝕜 f = spectrum 𝕜 ↑f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : PiLp p β
i : ι
h : 1 ≤ p.toReal
cancel : p.toReal * (1 / p.toReal) = 1
⊢ edist (x.ofLp i) (y.ofLp i) = (edist (x.ofLp i) (y.ofLp i) ^ p.toReal) ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : PiLp p β
i : ι
h : 1 ≤ p.toReal
cancel : p.toReal * (1 / p.toReal) = 1
⊢ (edist (x.ofLp i) (y.ofLp i) ^ p.toReal) ^ (1 / p.toReal) ≤
    (∑ i, edist (x.ofLp i) (y.ofLp i) ^ p.toReal) ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : PiLp p β
i : ι
⊢ edist (x.ofLp i) (y.ofLp i) ≤ edist x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
α : Type u_2
β : Type u_3
α' : Type u_5
β' : Type u_6
hp : Fact (1 ≤ p)
inst✝³ : PseudoEMetricSpace α
inst✝² : PseudoEMetricSpace β
inst✝¹ : PseudoEMetricSpace α'
inst✝ : PseudoEMetricSpace β'
f : α → α'
hf : Isometry f
g : β → β'
hg : Isometry g
⊢ Isometry (WithLp.map p (Prod.map f g))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : WithLp p ((i : ι) → β i)
⊢ edist x.ofLp y.ofLp ≤ edist x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
F✝ : Type u_4
inst✝¹⁰ : NormedAddCommGroup F✝
inst✝⁹ : NormedSpace 𝕜' F✝
f✝ : E →SL[σ] F✝
σ' : 𝕜' →+* 𝕜
inst✝⁸ : RingHomInvPair σ σ'
inst✝⁷ : RingHomIsometric σ
inst✝⁶ : RingHomIsometric σ'
inst✝⁵ : CompleteSpace F✝
inst✝⁴ : CompleteSpace E
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (↑f).range G
inst✝ : CompleteSpace ↥G
hker : (↑f).ker = ⊥
⊢ (↑(f.coprod G.subtypeL)).ker = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
F✝ : Type u_4
inst✝¹⁰ : NormedAddCommGroup F✝
inst✝⁹ : NormedSpace 𝕜' F✝
f✝ : E →SL[σ] F✝
σ' : 𝕜' →+* 𝕜
inst✝⁸ : RingHomInvPair σ σ'
inst✝⁷ : RingHomIsometric σ
inst✝⁶ : RingHomIsometric σ'
inst✝⁵ : CompleteSpace F✝
inst✝⁴ : CompleteSpace E
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (↑f).range G
inst✝ : CompleteSpace ↥G
hker : (↑f).ker = ⊥
⊢ (↑(f.coprod G.subtypeL)).range = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
α' : Type u_5
β' : Type u_6
hp : Fact (1 ≤ p)
inst✝⁴ : PseudoEMetricSpace α
inst✝³ : PseudoEMetricSpace β
inst✝² : PseudoEMetricSpace γ
inst✝¹ : PseudoEMetricSpace α'
inst✝ : PseudoEMetricSpace β'
x✝¹ x✝ : WithLp p (α × β)
⊢ edist ((WithLp.congr p (Equiv.prodComm α β)).toFun x✝¹) ((WithLp.congr p (Equiv.prodComm α β)).toFun x✝) =
    edist x✝¹ x✝

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
α' : Type u_5
β' : Type u_6
hp : Fact (1 ≤ p)
inst✝⁴ : PseudoEMetricSpace α
inst✝³ : PseudoEMetricSpace β
inst✝² : PseudoEMetricSpace γ
inst✝¹ : PseudoEMetricSpace α'
inst✝ : PseudoEMetricSpace β'
x✝¹ x✝ : WithLp p (WithLp p (α × β) × γ)
⊢ edist ((fun x ↦ WithLp.toLp p (x.fst.fst, WithLp.toLp p (x.fst.snd, x.snd))) x✝¹)
      ((fun x ↦ WithLp.toLp p (x.fst.fst, WithLp.toLp p (x.fst.snd, x.snd))) x✝) =
    edist x✝¹ x✝

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : CompleteSpace E
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (↑f).range G
inst✝ : CompleteSpace ↥G
hker : (↑f).ker = ⊥
⊢ (↑f).range = Submodule.map (↑↑(f.coprodSubtypeLEquivOfIsCompl h hker)) (⊤.prod ⊥)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
α' : Type u_5
β' : Type u_6
hp : Fact (1 ≤ p)
inst✝⁵ : PseudoEMetricSpace α
inst✝⁴ : PseudoEMetricSpace β
inst✝³ : PseudoEMetricSpace γ
inst✝² : PseudoEMetricSpace α'
inst✝¹ : PseudoEMetricSpace β'
inst✝ : Unique β
x y : WithLp p (α × β)
⊢ edist x.fst y.fst = edist x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : WithLp p ((i : ι) → β i)
h : 1 ≤ p.toReal
pos : 0 < p.toReal
nonneg : 0 ≤ 1 / p.toReal
cancel : p.toReal * (1 / p.toReal) = 1
⊢ (∑ i, edist (x.ofLp i) (y.ofLp i) ^ p.toReal) ^ (1 / p.toReal) ≤
    (∑ _i, edist x.ofLp y.ofLp ^ p.toReal) ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (↑f).range G
hG : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑G
hker : (↑f).ker = ⊥
⊢ IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(↑f).range

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : WithLp p ((i : ι) → β i)
h : 1 ≤ p.toReal
pos : 0 < p.toReal
nonneg : 0 ≤ 1 / p.toReal
cancel : p.toReal * (1 / p.toReal) = 1
⊢ (∑ _i, edist x.ofLp y.ofLp ^ p.toReal) ^ (1 / p.toReal) = ↑(↑(Fintype.card ι) ^ (1 / p.toReal)) * edist x.ofLp y.ofLp

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
⊢ AntilipschitzWith (↑(Fintype.card ι) ^ (1 / p).toReal) ofLp

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
⊢ 𝓤 (PiLp p β) = 𝓤 (WithLp p ((i : ι) → β i))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
inst✝² : Fintype ι
p : ℝ≥0∞
inst✝¹ : Fact (1 ≤ p)
β : ι → Type u_5
inst✝ : (i : ι) → PseudoMetricSpace (β i)
hp : p ≠ ∞
x y : PiLp p β
⊢ ↑(nndist x y) = ↑((∑ i, nndist (x.ofLp i) (y.ofLp i) ^ p.toReal) ^ (1 / p.toReal))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
inst✝¹ : Fintype ι
β : ι → Type u_5
inst✝ : (i : ι) → PseudoMetricSpace (β i)
x y : PiLp ∞ β
⊢ ↑(nndist x y) = ↑(⨆ i, nndist (x.ofLp i) (y.ofLp i))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoMetricSpace (β i)
x y : PiLp p β
i : ι
⊢ nndist (x.ofLp i) (y.ofLp i) ≤ nndist x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
x y : WithLp ∞ ((i : ι) → β i)
⊢ edist x.ofLp y.ofLp ≤ edist x y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
x y : WithLp ∞ ((i : ι) → β i)
⊢ edist x y ≤ edist x.ofLp y.ofLp

Tactic:
simpa only [ENNReal.div_top, ENNReal.toReal_zero, NNReal.rpow_zero, ENNReal.coe_one,
      one_mul] using antilipschitzWith_ofLp ∞ β x y

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
x y : WithLp ∞ ((i : ι) → β i)
⊢ edist x y ≤ edist x.ofLp y.ofLp

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
g : E →ₗ[𝕜] F
hg : IsClosed[instTopologicalSpaceProd] ↑g.graph
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ⇑g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp p β
h : 1 ≤ p.toReal
⊢ p ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp p β
⊢ dist x y = ‖-x + y‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x : PiLp p β
i : ι
⊢ ‖x.ofLp i‖ₑ ≤ ‖x‖ₑ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x : PiLp p β
i : ι
⊢ ‖x.ofLp i‖₊ ≤ ‖x‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x : PiLp p β
i : ι
⊢ ‖x.ofLp i‖ ≤ ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
g : E →ₗ[𝕜] F
hg : ∀ (u : ℕ → E) (x : E) (y : F), Tendsto u atTop (𝓝 x) → Tendsto (⇑g ∘ u) atTop (𝓝 y) → y = g x
φ : ℕ → E × F
x : E
y : F
hφg : ∀ (n : ℕ), φ n ∈ ↑g.graph
hφ : Tendsto φ atTop (𝓝 (x, y))
⊢ ⇑g ∘ Prod.fst ∘ φ = Prod.snd ∘ φ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
inst✝² : Fintype ι
p : ℝ≥0∞
inst✝¹ : Fact (1 ≤ p)
β : ι → Type u_5
hp : p ≠ ∞
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
f : PiLp p β
⊢ ‖f‖₊ = (∑ i, ‖f.ofLp i‖₊ ^ p.toReal) ^ (1 / p.toReal)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
f : PiLp ∞ β
⊢ ‖f‖₊ = ⨆ i, ‖f.ofLp i‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
f : PiLp ∞ β
⊢ ‖f.ofLp‖₊ = ‖f‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
inst✝² : Fintype ι
p : ℝ≥0∞
inst✝¹ : Fact (1 ≤ p)
β : ι → Type u_5
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
n : ℕ
h : p = ↑n
f : PiLp p β
⊢ ‖f‖ = (∑ i, ‖f.ofLp i‖ ^ n) ^ (1 / ↑n)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x : PiLp 1 β
⊢ ‖x‖ = ∑ i, ‖x.ofLp i‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
g : E →ₗ[𝕜] F
hg : ∀ (u : ℕ → E) (x : E) (y : F), Tendsto u atTop (𝓝 x) → Tendsto (⇑g ∘ u) atTop (𝓝 y) → y = g x
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ⇑g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x : PiLp 1 β
⊢ ↑‖x‖₊ = ↑(∑ i, ‖x.ofLp i‖₊)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
g : E →ₗ[𝕜] F
hg : IsClosed[instTopologicalSpaceProd] ↑g.graph
⊢ ↑(ofIsClosedGraph hg) = g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
g : E →ₗ[𝕜] F
hg : ∀ (u : ℕ → E) (x : E) (y : F), Tendsto u atTop (𝓝 x) → Tendsto (⇑g ∘ u) atTop (𝓝 y) → y = g x
⊢ ↑(ofSeqClosedGraph hg) = g

Tactic:
by

Goal state after: