Theorem: ContinuousLinearMap.norm_toSpanSingleton
File: Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean

tactic: refine opNorm_eq_of_bounds (norm_nonneg _) (fun x => ?_) fun N _ h => ?_
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
𝕜' : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
x : E
⊢ ‖toSpanSingleton 𝕜 x‖ = ‖x‖
state_after: case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
𝕜' : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
x✝ : E
x : 𝕜
⊢ ‖(toSpanSingleton 𝕜 x✝) x‖ ≤ ‖x✝‖ * ‖x‖

case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
𝕜' : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
x : E
N : ℝ
x✝ : N ≥ 0
h : ∀ (x_1 : 𝕜), ‖(toSpanSingleton 𝕜 x) x_1‖ ≤ N * ‖x_1‖
⊢ ‖x‖ ≤ N

tactic: rw [toSpanSingleton_apply, norm_smul, mul_comm]
state_before: case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
𝕜' : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
x✝ : E
x : 𝕜
⊢ ‖(toSpanSingleton 𝕜 x✝) x‖ ≤ ‖x✝‖ * ‖x‖
state_after: no goals

tactic: specialize h 1
state_before: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
𝕜' : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
x : E
N : ℝ
x✝ : N ≥ 0
h : ∀ (x_1 : 𝕜), ‖(toSpanSingleton 𝕜 x) x_1‖ ≤ N * ‖x_1‖
⊢ ‖x‖ ≤ N
state_after: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
𝕜' : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
x : E
N : ℝ
x✝ : N ≥ 0
h : ‖(toSpanSingleton 𝕜 x) 1‖ ≤ N * ‖1‖
⊢ ‖x‖ ≤ N

tactic: rw [toSpanSingleton_apply, norm_smul, mul_comm] at h
state_before: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
𝕜' : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
x : E
N : ℝ
x✝ : N ≥ 0
h : ‖(toSpanSingleton 𝕜 x) 1‖ ≤ N * ‖1‖
⊢ ‖x‖ ≤ N
state_after: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
𝕜' : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
x : E
N : ℝ
x✝ : N ≥ 0
h : ‖x‖ * ‖1‖ ≤ N * ‖1‖
⊢ ‖x‖ ≤ N

tactic: exact (mul_le_mul_right (by simp)).mp h
state_before: case refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
𝕜' : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
x : E
N : ℝ
x✝ : N ≥ 0
h : ‖x‖ * ‖1‖ ≤ N * ‖1‖
⊢ ‖x‖ ≤ N
state_after: no goals

tactic: simp
state_before: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
𝕜' : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' E
inst✝ : IsScalarTower 𝕜 𝕜' E
x : E
N : ℝ
x✝ : N ≥ 0
h : ‖x‖ * ‖1‖ ≤ N * ‖1‖
⊢ 0 < ‖1‖
state_after: no goals