M : Type u_1
N : Type ?u.81421
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
S : AddSubgroup M
x : M ⧸ S
⊢ sInf (norm '' {m | ↑m = -x}) = sInf (norm '' {m | ↑m = x})
M : Type u_1
N : Type ?u.81421
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
S : AddSubgroup M
x : M ⧸ S
⊢ ‖-x‖ = ‖x‖
simp only [AddSubgroup.quotient_norm_eq]
case e_a.h
M : Type u_1
N : Type ?u.81421
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
S : AddSubgroup M
x : M ⧸ S
r : ℝ
⊢ r ∈ norm '' {m | ↑m = -x} ↔ r ∈ norm '' {m | ↑m = x}
M : Type u_1
N : Type ?u.81421
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
S : AddSubgroup M
x : M ⧸ S
⊢ sInf (norm '' {m | ↑m = -x}) = sInf (norm '' {m | ↑m = x})
congr 1 with r
no goals
case e_a.h
M : Type u_1
N : Type ?u.81421
inst✝¹ : SeminormedAddCommGroup M
inst✝ : SeminormedAddCommGroup N
S : AddSubgroup M
x : M ⧸ S
r : ℝ
⊢ r ∈ norm '' {m | ↑m = -x} ↔ r ∈ norm '' {m | ↑m = x}
constructor <;> { rintro ⟨m, hm, rfl⟩; use -m; simpa [neg_eq_iff_eq_neg] using hm }