mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Module.ClosedSubmodule

Goal state before:
pp: ι : Sort u_1
R : Type u_2
M : Type u_3
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : Module R M
x : M
f : ι → ClosedSubmodule R M
⊢ x ∈ ⨅ i, f i ↔ ∀ (i : ι), x ∈ f i

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : Module R M
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
s : ClosedSubmodule R M
⊢ (↑s).closure = s

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : Module R M
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
s : Submodule R M
hs : IsClosed s.carrier
⊢ s.closure = { toSubmodule := s, isClosed' := hs }

Tactic:
by

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

Goal state before:
pp: R : Type u_2
N : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
s : ClosedSubmodule R N
⊢ (↑s).closure = s

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
inst✝² : Module R M
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
s : ClosedSubmodule R M
⊢ ↑(map (ContinuousLinearMap.id R M) s) = ↑s

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : Semiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : Module R M
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
f : M →L[R] N
s : ClosedSubmodule R M
t : ClosedSubmodule R N
⊢ map f s ≤ t ↔ s ≤ comap f t

Tactic:
by

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

Goal state before:
pp: R : Type u_2
N : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
s t : ClosedSubmodule R N
⊢ ↑(s ⊔ t) = closure (↑s ⊔ ↑t).carrier

Tactic:
by

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

Goal state before:
pp: ι : Sort u_1
R : Type u_2
N : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
f : ι → ClosedSubmodule R N
⊢ ↑(⨆ i, f i) = ↑(⨆ i, ↑(f i)).closure

Tactic:
by

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

Goal state before:
pp: R : Type u_2
N : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
S : Set (ClosedSubmodule R N)
⊢ ↑(sSup S) = closure (⨆ s ∈ S, ↑s).carrier

Tactic:
by

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

Goal state before:
pp: ι : Sort u_1
R : Type u_2
N : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
f : ι → ClosedSubmodule R N
⊢ ↑(⨆ i, f i) = closure (⨆ i, ↑(f i)).carrier

Tactic:
by

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

Goal state before:
pp: ι : Sort u_1
R : Type u_2
N : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
x : N
f : ι → ClosedSubmodule R N
⊢ x ∈ ⨆ i, f i ↔ x ∈ closure (⨆ i, ↑(f i)).carrier

Tactic:
by

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

Goal state before:
pp: ι : Sort u_1
R : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
inst✝¹¹ : Semiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : TopologicalSpace M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : TopologicalSpace N
inst✝⁵ : Module R N
inst✝⁴ : AddCommMonoid O
inst✝³ : TopologicalSpace O
inst✝² : Module R O
inst✝¹ : ContinuousAdd N
inst✝ : ContinuousConstSMul R N
f : M →L[R] N
s t : ClosedSubmodule R N
x : N
x✝ : Set (ClosedSubmodule R N)
⊢ IsLUB x✝ (sSup x✝)

Tactic:
by

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

Goal state before:
pp: ι : Sort u_1
R : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : TopologicalSpace N
inst✝³ : Module R N
inst✝² : AddCommMonoid O
inst✝¹ : TopologicalSpace O
inst✝ : Module R O
f : M ≃L[R] N
s : ClosedSubmodule R M
⊢ IsClosed (Submodule.map ↑↑f ↑s).carrier

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → X → ℝ≥0∞ → X
radius : ℕ → X → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → 0 < radius n x δ
HB : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → radius n x δ ≤ B (n + 1)
Hball : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → closedEBall (center n x δ) (radius n x δ) ⊆ closedEBall x δ ∩ f n
x : X
ε : ℝ≥0∞
εpos : 0 < ε
F : ℕ → X × ℝ≥0∞ := fun n ↦ Nat.recOn n (x, min ε (B 0)) fun n p ↦ (center n p.1 p.2, radius n p.1 p.2)
c : ℕ → X := fun n ↦ (F n).1
r : ℕ → ℝ≥0∞ := fun n ↦ (F n).2
rpos : ∀ (n : ℕ), 0 < r n
r0 : ∀ (n : ℕ), r n ≠ 0
rB : ∀ (n : ℕ), r n ≤ B n
incl : ∀ (n : ℕ), closedEBall (c (n + 1)) (r (n + 1)) ⊆ closedEBall (c n) (r n) ∩ f n
cdist : ∀ (n : ℕ), edist (c n) (c (n + 1)) ≤ B n
this : CauchySeq c
y : X
ylim : Tendsto c atTop (nhds y)
I : ∀ (n m : ℕ), m ≥ n → closedEBall (c m) (r m) ⊆ closedEBall (c n) (r n)
⊢ ∀ (n : ℕ), y ∈ closedEBall (c n) (r n)

Tactic:
by

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

Goal state before:
pp: ι : Sort u_1
R : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : TopologicalSpace N
inst✝³ : Module R N
inst✝² : AddCommMonoid O
inst✝¹ : TopologicalSpace O
inst✝ : Module R O
f : M ≃L[R] N
t : ClosedSubmodule R N
⊢ IsClosed (Submodule.map ↑↑f.symm ↑t).carrier

Tactic:
by

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

Goal state before:
pp: ι : Sort u_1
R : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : TopologicalSpace N
inst✝³ : Module R N
inst✝² : AddCommMonoid O
inst✝¹ : TopologicalSpace O
inst✝ : Module R O
f : M ≃L[R] N
⊢ LeftInverse (fun t ↦ { toSubmodule := Submodule.map ↑↑f.symm ↑t, isClosed' := ⋯ }) fun s ↦
    { toSubmodule := Submodule.map ↑↑f ↑s, isClosed' := ⋯ }

Tactic:
by

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

Goal state before:
pp: ι : Sort u_1
R : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
inst✝⁹ : Semiring R
inst✝⁸ : AddCommMonoid M
inst✝⁷ : TopologicalSpace M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid N
inst✝⁴ : TopologicalSpace N
inst✝³ : Module R N
inst✝² : AddCommMonoid O
inst✝¹ : TopologicalSpace O
inst✝ : Module R O
f : M ≃L[R] N
⊢ RightInverse (fun t ↦ { toSubmodule := Submodule.map ↑↑f.symm ↑t, isClosed' := ⋯ }) fun s ↦
    { toSubmodule := Submodule.map ↑↑f ↑s, isClosed' := ⋯ }

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
inst✝ : Module R N
f : M ≃L[R] N
s : ClosedSubmodule R M
x : M
⊢ f x ∈ (mapEquiv f) s ↔ x ∈ s

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : Semiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : TopologicalSpace M
inst✝⁵ : Module R M
inst✝⁴ : AddCommMonoid N
inst✝³ : TopologicalSpace N
inst✝² : Module R N
f : M ≃L[R] N
inst✝¹ : T1Space M
inst✝ : T1Space N
⊢ (mapEquiv f) ⊥ = ⊥

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
inst✝ : Module R N
f : M ≃L[R] N
⊢ (mapEquiv f) ⊤ = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
a : Finset ι
H : ∀ (m : Finset ι), a ⊆ m → ∀ (n : Finset ι), a ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε ^ 2 / 2
s₁ : Finset ι
hs₁ : a ⊆ s₁
s₂ : Finset ι
hs₂ : a ⊆ s₂
has : a ⊆ s₁ ⊓ s₂
⊢ ∑ x ∈ s₁ \ s₂, ‖f x‖ ^ 2 < ε ^ 2 / 2

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : TopologicalSpace N
inst✝ : Module R N
f : M ≃L[R] N
s t : ClosedSubmodule R M
⊢ (mapEquiv f) (s ⊓ t) = (mapEquiv f) s ⊓ (mapEquiv f) t

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
⊢ BaireSpace X

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Semiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : TopologicalSpace M
inst✝⁷ : Module R M
inst✝⁶ : AddCommMonoid N
inst✝⁵ : TopologicalSpace N
inst✝⁴ : Module R N
f : M ≃L[R] N
inst✝³ : ContinuousAdd N
inst✝² : ContinuousConstSMul R N
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
s : Submodule R M
⊢ (Submodule.map (↑↑f) s).closure = (mapEquiv f) s.closure

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Semiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : TopologicalSpace M
inst✝⁷ : Module R M
inst✝⁶ : AddCommMonoid N
inst✝⁵ : TopologicalSpace N
inst✝⁴ : Module R N
inst✝³ : ContinuousAdd N
inst✝² : ContinuousConstSMul R N
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
f : M ≃L[R] N
s t : ClosedSubmodule R M
x : N
⊢ ↑↑f = ↑↑f

Tactic:
by

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

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Semiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : TopologicalSpace M
inst✝⁷ : Module R M
inst✝⁶ : AddCommMonoid N
inst✝⁵ : TopologicalSpace N
inst✝⁴ : Module R N
inst✝³ : ContinuousAdd N
inst✝² : ContinuousConstSMul R N
inst✝¹ : ContinuousAdd M
inst✝ : ContinuousConstSMul R M
f : M ≃L[R] N
s t : ClosedSubmodule R M
⊢ (mapEquiv f) (s ⊔ t) = (mapEquiv f) s ⊔ (mapEquiv f) t

Tactic:
by

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

Goal state before:
pp: ι : Sort u_1
R : Type u_2
M : Type u_3
N : Type u_4
O : Type u_5
inst✝¹⁴ : Semiring R
inst✝¹³ : AddCommMonoid M
inst✝¹² : TopologicalSpace M
inst✝¹¹ : Module R M
inst✝¹⁰ : AddCommMonoid N
inst✝⁹ : TopologicalSpace N
inst✝⁸ : Module R N
inst✝⁷ : AddCommMonoid O
inst✝⁶ : TopologicalSpace O
inst✝⁵ : Module R O
𝕜 : Type u_6
H : Type u_7
inst✝⁴ : Semiring 𝕜
inst✝³ : AddCommMonoid H
inst✝² : UniformSpace H
inst✝¹ : Module 𝕜 H
inst✝ : CompleteSpace H
K : ClosedSubmodule 𝕜 H
⊢ CompleteSpace ↥K

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ⊆ m → ∀ (n : Finset ι), N ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
a : Finset ι
H : ∀ (m : Finset ι), a ⊆ m → ∀ (n : Finset ι), a ⊆ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε ^ 2 / 2
s₁ : Finset ι
hs₁ : a ⊆ s₁
s₂ : Finset ι
hs₂ : a ⊆ s₂
has : a ⊆ s₁ ⊓ s₂
Hs₁ : ∑ x ∈ s₁ \ s₂, ‖f x‖ ^ 2 < ε ^ 2 / 2
⊢ ∑ x ∈ s₂ \ s₁, ‖f x‖ ^ 2 < ε ^ 2 / 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
⊢ (Summable fun i ↦ (V i) (f i)) ↔ Summable fun i ↦ ‖f i‖ ^ 2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
V : ι → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
v : Π₀ (i : ι), ↥(V i)
hv : ((DFinsupp.lsum ℕ) fun i ↦ (V i).subtype) v = 0
i : ι
this : ⟪↑(v i), ↑(v i)⟫ = 0
⊢ ↑(v i) = ↑(0 i)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
V : ι → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
v : Π₀ (i : ι), ↥(V i)
hv : ((DFinsupp.lsum ℕ) fun i ↦ (V i).subtype) v = 0
i : ι
⊢ ⟪↑(v i), ↑(v i)⟫ = ⟪↑(v i), ((DFinsupp.lsum ℕ) fun i ↦ (V i).subtype) v⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
V : ι → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
v : Π₀ (i : ι), ↥(V i)
hv : ((DFinsupp.lsum ℕ) fun i ↦ (V i).subtype) v = 0
i : ι
⊢ ⟪↑(v i), ((DFinsupp.lsum ℕ) fun i ↦ (V i).subtype) v⟫ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
V : ι → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
⊢ iSupIndep V

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
hV_sum : IsInternal fun i ↦ V i
α : ι → Type u_6
v_family : (i : ι) → Basis (α i) 𝕜 ↥(V i)
hv_family : ∀ (i : ι), Orthonormal 𝕜 ⇑(v_family i)
⊢ Orthonormal 𝕜 ⇑(hV_sum.collectedBasis v_family)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
a✝ b✝ : E
hx : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
hy : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
u : E
hu : u ∈ K
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
this : ∀ (x : K), ∃ a b, a • 1 + b • I = x
⊢ Submodule.span ℝ {1, I} = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x : K
⊢ re x • 1 + im x • I = x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
⊢ Submodule.span ℝ {1, I} = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
c : 𝕜
x : E
hx : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
u : E
hu : u ∈ K
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
⊢ Module.rank ℝ K = Module.rank ℝ ↥(Submodule.span ℝ {1, I})

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
⊢ Module.rank ℝ ↥(Submodule.span ℝ {1, I}) ≤ ↑(#{1, I})

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
⊢ Submodule.span ℝ ↑{1, I} = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
v : E
⊢ v ∈ Kᗮ ↔ ∀ u ∈ K, ⟪v, u⟫ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝³ : RCLike K
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace K E
inst✝ : FiniteDimensional K E
⊢ ProperSpace E

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
⊢ ‖reCLM‖ = 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
u v : E
hu : u ∈ K
hv : v ∈ Kᗮ
⊢ ⟪v, u⟫ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
u v : E
⊢ v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪u, v⟫ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
u v : E
⊢ v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪v, u⟫ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
x y : E
h : ∀ (v : ↥K), ⟪x, ↑v⟫ = ⟪y, ↑v⟫
⊢ x - y ∈ Kᗮ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
x y : E
h : ∀ (v : ↥K), ⟪↑v, x⟫ = ⟪↑v, y⟫
⊢ x - y ∈ Kᗮ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
⊢ (starRingEnd 𝕜) ⟪T x, y⟫ = ⟪T y, x⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
⊢ K ⊓ Kᗮ = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
⊢ Disjoint K Kᗮ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
⊢ Kᗮ = ⨅ v, (↑((innerSL 𝕜) ↑v)).ker

Tactic:
ext
  simpa using mem_orthogonal _ _

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
⊢ Kᗮ = ⨅ v, (↑((innerSL 𝕜) ↑v)).ker

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
⊢ (T + S).IsSymmetric

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
⊢ IsClosed ↑Kᗮ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_3
T : ι → E →ₗ[𝕜] E
s : Finset ι
hT : ∀ i ∈ s, (T i).IsSymmetric
x✝¹ x✝ : E
⊢ ⟪(∑ i ∈ s, T i) x✝¹, x✝⟫ = ⟪x✝¹, (∑ i ∈ s, T i) x✝⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
f : E →ₗᵢ[𝕜] F
⊢ map f.toLinearMap Kᗮ = (map f.toLinearMap K)ᗮ ⊓ f.range

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T S : E →ₗ[𝕜] E
hT : T.IsSymmetric
hS : S.IsSymmetric
⊢ (T - S).IsSymmetric

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
c : 𝕜
hc : (starRingEnd 𝕜) c = c
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
⊢ (c • T).IsSymmetric

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
n : ℕ
x y : E
⊢ ⟪↑n x, y⟫ = ⟪x, ↑n y⟫

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
n : ℤ
x y : E
⊢ ⟪↑n x, y⟫ = ⟪x, ↑n y⟫

Tactic:
by

Goal state after: