mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: E : Type u_4
p : ℝ≥0∞
inst✝¹ : NormedAddCommGroup E
β : Type u_7
inst✝ : MeasurableSpace β
μb : Measure β
x✝ : ↥(Lp E p μb)
⊢ ↑↑((compMeasurePreserving id ⋯) x✝) =ᵐ[μb] ↑↑((AddMonoidHom.id ↥(Lp E p μb)) x✝)

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: E : Type u_4
p : ℝ≥0∞
inst✝¹ : NormedAddCommGroup E
β : Type u_7
inst✝ : MeasurableSpace β
μb : Measure β
g : ↥(Lp E p μb)
⊢ (compMeasurePreserving id ⋯) g = g

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inl
ι : Type u
s : Finset ι
w f : ι → ℝ≥0∞
hp : 1 ≤ 1
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - 1⁻¹) * (∑ i ∈ s, w i * f i ^ 1) ^ 1⁻¹

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case zero
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → α
hf : MeasurePreserving f μ μ
x✝ : ↥(Lp E p μ)
⊢ (⇑(compMeasurePreserving f hf))^[0] x✝ = (compMeasurePreserving f^[0] ⋯) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
f : α → E
h_Lp : MemLp f p μ
L : E →L[𝕜] F
⊢ L 0 = 0

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜'
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝⁴ : RingHomIsometric σ
𝕜'' : Type u_8
inst✝³ : NormedRing 𝕜''
inst✝² : Module 𝕜'' F
inst✝¹ : IsBoundedSMul 𝕜'' F
inst✝ : SMulCommClass 𝕜' 𝕜'' F
c : 𝕜''
L : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ (c • L).compLp f = c • L.compLp f

Tactic:
by
  ext1
  grw [Lp.coeFn_smul, coeFn_compLp', coeFn_compLp']
  rfl

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case ha
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ⋯
S : ℕ → Set α := ⋯
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := ⋯
hs : s = limsup S atTop
hμs : μ s = 0
x : α
ε : ℝ≥0∞
hε : ε > 0
N k : ℕ
hk_lt_ε : 2 * 2⁻¹ ^ k < ε
n : ℕ
hNx : edist (f (ns n) x) (g x) < 2⁻¹ ^ n
hn_ge : n ≥ max N (k - 1)
⊢ 1 ≤ 2

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case h₁
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
x : α
ε : ℝ≥0∞
hε : ε > 0
N k : ℕ
hk_lt_ε : 2 * 2⁻¹ ^ k < ε
n : ℕ
hNx : edist (f (ns n) x) (g x) < 2⁻¹ ^ n
hn_ge : n ≥ max N (k - 1)
⊢ 2 ^ n * 2 ^ 1 = ∞ → (2 ^ k)⁻¹ ≠ 0

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case h₂
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
x : α
ε : ℝ≥0∞
hε : ε > 0
N k : ℕ
hk_lt_ε : 2 * 2⁻¹ ^ k < ε
n : ℕ
hNx : edist (f (ns n) x) (g x) < 2⁻¹ ^ n
hn_ge : n ≥ max N (k - 1)
⊢ (2 ^ k)⁻¹ = ∞ → 2 ^ n * 2 ^ 1 ≠ 0

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case h₁
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
x : α
ε : ℝ≥0∞
hε : ε > 0
N k : ℕ
hk_lt_ε : 2 * 2⁻¹ ^ k < ε
n : ℕ
hNx : edist (f (ns n) x) (g x) < 2⁻¹ ^ n
hn_ge : n ≥ max N (k - 1)
⊢ (2 ^ k)⁻¹ * 2 ^ 1 = ∞ → 2 ^ n ≠ 0

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: case h₂
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
x : α
ε : ℝ≥0∞
hε : ε > 0
N k : ℕ
hk_lt_ε : 2 * 2⁻¹ ^ k < ε
n : ℕ
hNx : edist (f (ns n) x) (g x) < 2⁻¹ ^ n
hn_ge : n ≥ max N (k - 1)
⊢ 2 ^ n = ∞ → (2 ^ k)⁻¹ * 2 ^ 1 ≠ 0

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
ν : Measure α
c : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
hc : c ≠ ∞
h : μ ≤ c • ν
⊢ 0 ≤ c.toReal

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ↥(Lp E p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ε = ∞ → μ {x | ‖↑↑f x‖ₑ = ∞} = 0

Tactic:
simp

Goal state after:
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
x✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }
f : α ↪ γ
g : β ↪ γ
disj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)
a₁ : α
b₂ : β
f_eq : f a₁ = g b₂
⊢ (f, g).2 b₂ = g b₂

Tactic:
simp

Goal state after:
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : Unique α
x✝ : β
⊢ (fun f ↦ f default) ((fun x ↦ { toFun := fun x_1 ↦ x, inj' := ⋯ }) x✝) = x✝

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hp : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N
hp_top : ¬p = ∞
⊢ ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) atTop (𝓝 l)

Tactic:
have hp1 : 1 ≤ p.toReal := by
    rw [← ENNReal.ofReal_le_iff_le_toReal hp_top, ENNReal.ofReal_one]
    exact hp

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hp : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N
hp_top : ¬p = ∞
hp1 : 1 ≤ p.toReal
⊢ ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n ↦ f n x) atTop (𝓝 l)
Module: Mathlib.Topology.Algebra.Module.Multilinear.Basic

Goal state before:
pp: R : Type u
ι : Type v
n : ℕ
M : Fin n.succ → Type w
M₁ : ι → Type w₁
M₁' : ι → Type w₁'
M₂ : Type w₂
M₃ : Type w₃
M₄ : Type w₄
inst✝¹⁸ : Semiring R
inst✝¹⁷ : (i : Fin n.succ) → AddCommMonoid (M i)
inst✝¹⁶ : (i : ι) → AddCommMonoid (M₁ i)
inst✝¹⁵ : (i : ι) → AddCommMonoid (M₁' i)
inst✝¹⁴ : AddCommMonoid M₂
inst✝¹³ : AddCommMonoid M₃
inst✝¹² : AddCommMonoid M₄
inst✝¹¹ : (i : Fin n.succ) → Module R (M i)
inst✝¹⁰ : (i : ι) → Module R (M₁ i)
inst✝⁹ : (i : ι) → Module R (M₁' i)
inst✝⁸ : Module R M₂
inst✝⁷ : Module R M₃
inst✝⁶ : Module R M₄
inst✝⁵ : (i : Fin n.succ) → TopologicalSpace (M i)
inst✝⁴ : (i : ι) → TopologicalSpace (M₁ i)
inst✝³ : (i : ι) → TopologicalSpace (M₁' i)
inst✝² : TopologicalSpace M₂
inst✝¹ : TopologicalSpace M₃
inst✝ : TopologicalSpace M₄
f f' : ContinuousMultilinearMap R M₁ M₂
ι' : Type u_1
e : ι ≃ ι'
x✝¹ : ContinuousMultilinearMap R (fun x ↦ M₂) M₃
x✝ : ι → M₂
⊢ (domDomCongr e.symm (domDomCongr e x✝¹)) x✝ = x✝¹ x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Basic

Goal state before:
pp: R : Type u
ι : Type v
n : ℕ
M : Fin n.succ → Type w
M₁ : ι → Type w₁
M₁' : ι → Type w₁'
M₂ : Type w₂
M₃ : Type w₃
M₄ : Type w₄
inst✝¹⁸ : Semiring R
inst✝¹⁷ : (i : Fin n.succ) → AddCommMonoid (M i)
inst✝¹⁶ : (i : ι) → AddCommMonoid (M₁ i)
inst✝¹⁵ : (i : ι) → AddCommMonoid (M₁' i)
inst✝¹⁴ : AddCommMonoid M₂
inst✝¹³ : AddCommMonoid M₃
inst✝¹² : AddCommMonoid M₄
inst✝¹¹ : (i : Fin n.succ) → Module R (M i)
inst✝¹⁰ : (i : ι) → Module R (M₁ i)
inst✝⁹ : (i : ι) → Module R (M₁' i)
inst✝⁸ : Module R M₂
inst✝⁷ : Module R M₃
inst✝⁶ : Module R M₄
inst✝⁵ : (i : Fin n.succ) → TopologicalSpace (M i)
inst✝⁴ : (i : ι) → TopologicalSpace (M₁ i)
inst✝³ : (i : ι) → TopologicalSpace (M₁' i)
inst✝² : TopologicalSpace M₂
inst✝¹ : TopologicalSpace M₃
inst✝ : TopologicalSpace M₄
f f' : ContinuousMultilinearMap R M₁ M₂
ι' : Type u_1
e : ι ≃ ι'
x✝¹ : ContinuousMultilinearMap R (fun x ↦ M₂) M₃
x✝ : ι' → M₂
⊢ (domDomCongr e (domDomCongr e.symm x✝¹)) x✝ = x✝¹ x✝

Tactic:
simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
hp : 1 ≤ p
f : ℕ → α → E
hf : ∀ (n : ℕ), MemLp (f n) p μ
f_lim : α → E
h_lim_meas : AEStronglyMeasurable f_lim μ
h_tendsto : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - f_lim) p μ ≤ ε
N : ℕ
h_tendsto_1 : eLpNorm (f N - f_lim) p μ ≤ 1
h_add : f_lim = f_lim - f N + f N
⊢ f_lim - f N = -(f N - f_lim)

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
y : R
h : ‖y - ↑x‖ < ‖↑x⁻¹‖⁻¹
⊢ y = ↑(x.add (y - ↑x) h)

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
⊢ 2⁻¹ > 0

Tactic:
simp

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
t : R
ht : ‖t‖ < 2⁻¹
ht' : ‖t‖ < 1
this : ‖∑' (n : ℕ), t ^ n‖ ≤ ‖1‖ - 1 + (1 - ‖t‖)⁻¹
⊢ 0 < 2

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x✝ : E
⊢ ‖(toSpanSingletonLE 𝕜 𝕜 E) x✝‖ = ‖x✝‖

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝⁷ : SeminormedAddCommGroup E
inst✝⁶ : SeminormedAddCommGroup F
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : NontrivialTopology E
inst✝ : RingHomIsometric σ₁₂
f : E →ₛₗᵢ[σ₁₂] F
⊢ ∀ (x : E), ‖f.toContinuousLinearMap x‖ = 1 * ‖x‖

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_8
inst✝¹⁰ : SeminormedAddCommGroup E
inst✝⁹ : SeminormedAddCommGroup F
inst✝⁸ : SeminormedAddCommGroup G
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NontriviallyNormedField 𝕜₃
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜₂ F
inst✝² : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝ : RingHomIsometric σ₁₂
f : F →ₛₗᵢ[σ₂₃] G
g : E →SL[σ₁₂] F
x : E
⊢ ‖(f.toContinuousLinearMap ∘SL g) x‖ = ‖g x‖

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₁ : Type u_2
𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
inst✝¹¹ : SeminormedAddCommGroup E
inst✝¹⁰ : SeminormedAddCommGroup F
inst✝⁹ : SeminormedAddCommGroup G
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜₂
inst✝⁶ : NontriviallyNormedField 𝕜₃
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ : E →SL[σ₁₂] F
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : RingHomIsometric σ₁₃
a : F →ₛₗᵢ[σ₂₃] G
f g : E →SL[σ₁₂] F
⊢ a.toContinuousLinearMap ∘SL (f + g) = a.toContinuousLinearMap ∘SL f + a.toContinuousLinearMap ∘SL g

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₁ : Type u_2
𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
inst✝¹¹ : SeminormedAddCommGroup E
inst✝¹⁰ : SeminormedAddCommGroup F
inst✝⁹ : SeminormedAddCommGroup G
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜₂
inst✝⁶ : NontriviallyNormedField 𝕜₃
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
f✝ : E →SL[σ₁₂] F
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : RingHomIsometric σ₁₃
a : F →ₛₗᵢ[σ₂₃] G
c : 𝕜₂
f : E →SL[σ₁₂] F
⊢ a.toContinuousLinearMap ∘SL (c • f) = σ₂₃ c • a.toContinuousLinearMap ∘SL f

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁵ : NormedDivisionRing R
inst✝⁴ : NormedAlgebra 𝕜 R
inst✝³ : Module R E
inst✝² : NormSMulClass R E
inst✝¹ : IsScalarTower 𝕜 R E
inst✝ : Nontrivial E
⊢ ↑‖lsmul 𝕜 R‖₊ = ↑1

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁵ : NormedDivisionRing R
inst✝⁴ : NormedAlgebra 𝕜 R
inst✝³ : Module R E
inst✝² : NormSMulClass R E
inst✝¹ : IsScalarTower 𝕜 R E
inst✝ : Nontrivial E
a : R
⊢ ↑‖(lsmul 𝕜 R) a‖₊ = ↑‖a‖₊

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : Semiring 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : Module 𝕜 F
⊢ ∀ (x : E), ‖0 x‖ ≤ 0 * ‖x‖

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : Semiring 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : Module 𝕜 E
⊢ ∀ (x : E), ‖LinearMap.id x‖ ≤ 1 * ‖x‖

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Determinant

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : CommRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : ContinuousMul 𝕜
f : 𝕜 →L[𝕜] 𝕜
v : 𝕜
⊢ (f.smulRight v).det = f 1 * v

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Determinant

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : CommRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : ContinuousMul 𝕜
v : 𝕜
⊢ (ContinuousLinearMap.id 𝕜 𝕜) 1 * v = v

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Quotient

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submodule R M
x : M
⊢ S.mkQL x = S.mkQ x

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Quotient

Goal state before:
pp: R : Type u_1
R₂ : Type u_2
inst✝⁷ : Ring R
inst✝⁶ : Ring R₂
σ : R →+* R₂
M : Type u_3
M₂ : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : TopologicalSpace M₂
inst✝¹ : AddCommGroup M₂
inst✝ : Module R₂ M₂
S : Submodule R M
f : M →SL[σ] M₂
h : S ≤ (↑f).ker
x : M ⧸ S
⊢ (S.liftQL f h) x = (S.liftQ (↑f) h) x

Tactic:
simp

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace E
e : E ≃L[𝕜] F
⊢ ↑e ∈ range toContinuousLinearMap

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁸ : NormedField 𝕜
inst✝⁷ : (i : ι) → TopologicalSpace (E i)
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
S : Set (ContinuousMultilinearMap 𝕜 E F)
hS : IsVonNBounded 𝕜 S
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
U : Set F
hU : U ∈ 𝓝 0
c : 𝕜
hc : S ⊆ c • {x | MapsTo (⇑x) s U}
f : ContinuousMultilinearMap 𝕜 E F
⊢ f ∈ S → ∀ y ∈ s, f y ∈ c • U

Tactic:
hf

Goal state after:
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝⁸ : NormedField 𝕜
inst✝⁷ : (i : ι) → TopologicalSpace (E i)
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜 F
inst✝² : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
S : Set (ContinuousMultilinearMap 𝕜 E F)
hS : IsVonNBounded 𝕜 S
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
U : Set F
hU : U ∈ 𝓝 0
c : 𝕜
hc : S ⊆ c • {x | MapsTo (⇑x) s U}
f : ContinuousMultilinearMap 𝕜 E F
hf : f ∈ S
⊢ ∀ y ∈ s, f y ∈ c • U
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
E₁ : ι → Type u_5
inst✝³ : (i : ι) → TopologicalSpace (E₁ i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝ : (i : ι) → Module 𝕜 (E₁ i)
f : (i : ι) → E i →L[𝕜] E₁ i
x✝² x✝¹ : ContinuousMultilinearMap 𝕜 E₁ F
x✝ : (i : ι) → E i
⊢ ((x✝² + x✝¹).compContinuousLinearMap f) x✝ = (x✝².compContinuousLinearMap f + x✝¹.compContinuousLinearMap f) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
E₁ : ι → Type u_5
inst✝³ : (i : ι) → TopologicalSpace (E₁ i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝ : (i : ι) → Module 𝕜 (E₁ i)
f : (i : ι) → E i →L[𝕜] E₁ i
x✝² : 𝕜
x✝¹ : ContinuousMultilinearMap 𝕜 E₁ F
x✝ : (i : ι) → E i
⊢ ((x✝² • x✝¹).compContinuousLinearMap f) x✝ = ((RingHom.id 𝕜) x✝² • x✝¹.compContinuousLinearMap f) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
g : F →L[𝕜] G
x✝² x✝¹ : ContinuousMultilinearMap 𝕜 E F
x✝ : (i : ι) → E i
⊢ (g.compContinuousMultilinearMap (x✝² + x✝¹)) x✝ =
    (g.compContinuousMultilinearMap x✝² + g.compContinuousMultilinearMap x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
g : F →L[𝕜] G
x✝² : 𝕜
x✝¹ : ContinuousMultilinearMap 𝕜 E F
x✝ : (i : ι) → E i
⊢ (g.compContinuousMultilinearMap (x✝² • x✝¹)) x✝ = ((RingHom.id 𝕜) x✝² • g.compContinuousMultilinearMap x✝¹) x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
x✝³ x✝² : F →L[𝕜] G
x✝¹ : ContinuousMultilinearMap 𝕜 E F
x✝ : (i : ι) → E i
⊢ ({ toFun := (x✝³ + x✝²).compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } x✝¹) x✝ =
    (({ toFun := x✝³.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } +
          { toFun := x✝².compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ })
        x✝¹)
      x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
x✝³ : 𝕜
x✝² : F →L[𝕜] G
x✝¹ : ContinuousMultilinearMap 𝕜 E F
x✝ : (i : ι) → E i
⊢ ({ toFun := (x✝³ • x✝²).compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } x✝¹) x✝ =
    (((RingHom.id 𝕜) x✝³ • { toFun := x✝².compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }) x✝¹)
      x✝

Tactic:
simp

Goal state after: