Step |
Hyp |
Ref |
Expression |
1 |
|
ngpi.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
ngpi.n |
⊢ 𝑁 = ( norm ‘ 𝑊 ) |
3 |
|
ngpi.m |
⊢ − = ( -g ‘ 𝑊 ) |
4 |
|
ngpi.0 |
⊢ 0 = ( 0g ‘ 𝑊 ) |
5 |
|
ngpgrp |
⊢ ( 𝑊 ∈ NrmGrp → 𝑊 ∈ Grp ) |
6 |
1 2
|
nmf |
⊢ ( 𝑊 ∈ NrmGrp → 𝑁 : 𝑉 ⟶ ℝ ) |
7 |
1 2 4
|
nmeq0 |
⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ) → ( ( 𝑁 ‘ 𝑥 ) = 0 ↔ 𝑥 = 0 ) ) |
8 |
1 2 3
|
nmmtri |
⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) |
9 |
8
|
3expa |
⊢ ( ( ( 𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ) ∧ 𝑦 ∈ 𝑉 ) → ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) |
10 |
9
|
ralrimiva |
⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ) → ∀ 𝑦 ∈ 𝑉 ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) |
11 |
7 10
|
jca |
⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ) → ( ( ( 𝑁 ‘ 𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦 ∈ 𝑉 ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) |
12 |
11
|
ralrimiva |
⊢ ( 𝑊 ∈ NrmGrp → ∀ 𝑥 ∈ 𝑉 ( ( ( 𝑁 ‘ 𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦 ∈ 𝑉 ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) |
13 |
5 6 12
|
3jca |
⊢ ( 𝑊 ∈ NrmGrp → ( 𝑊 ∈ Grp ∧ 𝑁 : 𝑉 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑉 ( ( ( 𝑁 ‘ 𝑥 ) = 0 ↔ 𝑥 = 0 ) ∧ ∀ 𝑦 ∈ 𝑉 ( 𝑁 ‘ ( 𝑥 − 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) ) |