Theorem: AddCircle.norm_coe_mul
File: Mathlib/Analysis/Normed/Group/AddCircle.lean

Proof steps:
state_before: p x t : ℝ
⊢ ‖↑(t * x)‖ = |t| * ‖↑x‖
state_after: p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
⊢ ‖↑(t * x)‖ = |t| * ‖↑x‖
tactic: have aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b) := fun {a b c} h => by
    simp only [mem_zmultiples_iff] at h ⊢
    obtain ⟨n, rfl⟩ := h
    exact ⟨n, (mul_smul_comm n c b).symm⟩
state_before: p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
⊢ ‖↑(t * x)‖ = |t| * ‖↑x‖
state_after: case inl
p x : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
⊢ ‖↑(0 * x)‖ = |0| * ‖↑x‖

case inr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
⊢ ‖↑(t * x)‖ = |t| * ‖↑x‖
tactic: rcases eq_or_ne t 0 with (rfl | ht)
state_before: case inr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
⊢ ‖↑(t * x)‖ = |t| * ‖↑x‖
state_after: case inr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
⊢ ‖↑(t * x)‖ = |t| * ‖↑x‖
tactic: have ht' : |t| ≠ 0 := (not_congr abs_eq_zero).mpr ht
state_before: case inr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
⊢ ‖↑(t * x)‖ = |t| * ‖↑x‖
state_after: case inr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
⊢ sInf ((fun a => |a|) '' {m | ↑m = ↑(t * x)}) = |t| * sInf ((fun a => |a|) '' {m | ↑m = ↑x})
tactic: simp only [quotient_norm_eq, Real.norm_eq_abs]
state_before: case inr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
⊢ sInf ((fun a => |a|) '' {m | ↑m = ↑(t * x)}) = |t| * sInf ((fun a => |a|) '' {m | ↑m = ↑x})
state_after: case inr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
⊢ sInf ((fun a => |a|) '' {m | ↑m = ↑(t * x)}) = sInf (|t| • (fun a => |a|) '' {m | ↑m = ↑x})
tactic: conv_rhs => rw [← smul_eq_mul, ← Real.sInf_smul_of_nonneg (abs_nonneg t)]
state_before: case inr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
⊢ sInf ((fun a => |a|) '' {m | ↑m = ↑(t * x)}) = sInf (|t| • (fun a => |a|) '' {m | ↑m = ↑x})
state_after: case inr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
⊢ sInf ((fun a => |a|) '' {m | m - t * x ∈ zmultiples (t * p)}) =
    sInf (|t| • (fun a => |a|) '' {m | m - x ∈ zmultiples p})
tactic: simp only [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_iff_sub_mem]
state_before: case inr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
⊢ sInf ((fun a => |a|) '' {m | m - t * x ∈ zmultiples (t * p)}) =
    sInf (|t| • (fun a => |a|) '' {m | m - x ∈ zmultiples p})
state_after: case inr.e_a
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
⊢ (fun a => |a|) '' {m | m - t * x ∈ zmultiples (t * p)} = |t| • (fun a => |a|) '' {m | m - x ∈ zmultiples p}
tactic: congr 1
state_before: case inr.e_a
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
⊢ (fun a => |a|) '' {m | m - t * x ∈ zmultiples (t * p)} = |t| • (fun a => |a|) '' {m | m - x ∈ zmultiples p}
state_after: case inr.e_a.h
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ z ∈ (fun a => |a|) '' {m | m - t * x ∈ zmultiples (t * p)} ↔ z ∈ |t| • (fun a => |a|) '' {m | m - x ∈ zmultiples p}
tactic: ext z
state_before: case inr.e_a.h
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ z ∈ (fun a => |a|) '' {m | m - t * x ∈ zmultiples (t * p)} ↔ z ∈ |t| • (fun a => |a|) '' {m | m - x ∈ zmultiples p}
state_after: case inr.e_a.h
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ z ∈ (fun a => |a|) '' {m | m - t * x ∈ zmultiples (t * p)} ↔ |t|⁻¹ • z ∈ (fun a => |a|) '' {m | m - x ∈ zmultiples p}
tactic: rw [mem_smul_set_iff_inv_smul_mem₀ ht']
state_before: case inr.e_a.h
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ z ∈ (fun a => |a|) '' {m | m - t * x ∈ zmultiples (t * p)} ↔ |t|⁻¹ • z ∈ (fun a => |a|) '' {m | m - x ∈ zmultiples p}
state_after: case inr.e_a.h
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ (∃ y, y - t * x ∈ zmultiples (t * p) ∧ |y| = z) ↔ ∃ w, w - x ∈ zmultiples p ∧ |w| = |t|⁻¹ * z
tactic: show
    (∃ y, y - t * x ∈ zmultiples (t * p) ∧ |y| = z) ↔ ∃ w, w - x ∈ zmultiples p ∧ |w| = |t|⁻¹ * z
state_before: case inr.e_a.h
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ (∃ y, y - t * x ∈ zmultiples (t * p) ∧ |y| = z) ↔ ∃ w, w - x ∈ zmultiples p ∧ |w| = |t|⁻¹ * z
state_after: case inr.e_a.h.mp
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ (∃ y, y - t * x ∈ zmultiples (t * p) ∧ |y| = z) → ∃ w, w - x ∈ zmultiples p ∧ |w| = |t|⁻¹ * z

case inr.e_a.h.mpr
p x t : ℝ
aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)
ht : t ≠ 0
ht' : |t| ≠ 0
z : ℝ
⊢ (∃ w, w - x ∈ zmultiples p ∧ |w| = |t|⁻¹ * z) → ∃ y, y - t * x ∈ zmultiples (t * p) ∧ |y| = z
tactic: constructor