Theorem: IsCoercive.range_eq_top
File: Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean

Proof steps:
state_before: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
⊢ range (continuousLinearMapOfBilin B) = ⊤
state_after: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
⊢ range (continuousLinearMapOfBilin B) = ⊤
tactic: haveI := coercive.isClosed_range.completeSpace_coe
state_before: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
⊢ range (continuousLinearMapOfBilin B) = ⊤
state_after: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
⊢ (range (continuousLinearMapOfBilin B))ᗮᗮ = ⊤
tactic: rw [← (range B♯).orthogonal_orthogonal]
state_before: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
⊢ (range (continuousLinearMapOfBilin B))ᗮᗮ = ⊤
state_after: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
⊢ ∀ (x : V), x ∈ (range (continuousLinearMapOfBilin B))ᗮᗮ
tactic: rw [Submodule.eq_top_iff']
state_before: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
⊢ ∀ (x : V), x ∈ (range (continuousLinearMapOfBilin B))ᗮᗮ
state_after: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
v w : V
mem_w_orthogonal : w ∈ (range (continuousLinearMapOfBilin B))ᗮ
⊢ ⟪w, v⟫_ℝ = 0
tactic: intro v w mem_w_orthogonal
state_before: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
v w : V
mem_w_orthogonal : w ∈ (range (continuousLinearMapOfBilin B))ᗮ
⊢ ⟪w, v⟫_ℝ = 0
state_after: case intro.intro
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
v w : V
mem_w_orthogonal : w ∈ (range (continuousLinearMapOfBilin B))ᗮ
C : ℝ
C_pos : 0 < C
coercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u
⊢ ⟪w, v⟫_ℝ = 0
tactic: rcases coercive with ⟨C, C_pos, coercivity⟩
state_before: case intro.intro
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
this : CompleteSpace ↑↑(range (continuousLinearMapOfBilin B))
v : V
C : ℝ
C_pos : 0 < C
coercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u
mem_w_orthogonal : 0 ∈ (range (continuousLinearMapOfBilin B))ᗮ
⊢ ⟪0, v⟫_ℝ = 0
state_after: no goals
tactic: exact inner_zero_left _