Step |
Hyp |
Ref |
Expression |
1 |
|
nvz.1 |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
nvz.5 |
⊢ 𝑍 = ( 0vec ‘ 𝑈 ) |
3 |
|
nvz.6 |
⊢ 𝑁 = ( normCV ‘ 𝑈 ) |
4 |
|
eqid |
⊢ ( +𝑣 ‘ 𝑈 ) = ( +𝑣 ‘ 𝑈 ) |
5 |
|
eqid |
⊢ ( ·𝑠OLD ‘ 𝑈 ) = ( ·𝑠OLD ‘ 𝑈 ) |
6 |
1 4 5 2 3
|
nvi |
⊢ ( 𝑈 ∈ NrmCVec → ( 〈 ( +𝑣 ‘ 𝑈 ) , ( ·𝑠OLD ‘ 𝑈 ) 〉 ∈ CVecOLD ∧ 𝑁 : 𝑋 ⟶ ℝ ∧ ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 ( ·𝑠OLD ‘ 𝑈 ) 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 ( +𝑣 ‘ 𝑈 ) 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) ) |
7 |
6
|
simp3d |
⊢ ( 𝑈 ∈ NrmCVec → ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 ( ·𝑠OLD ‘ 𝑈 ) 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 ( +𝑣 ‘ 𝑈 ) 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) ) |
8 |
|
simp1 |
⊢ ( ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 ( ·𝑠OLD ‘ 𝑈 ) 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 ( +𝑣 ‘ 𝑈 ) 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) → ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ) |
9 |
8
|
ralimi |
⊢ ( ∀ 𝑥 ∈ 𝑋 ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ∧ ∀ 𝑦 ∈ ℂ ( 𝑁 ‘ ( 𝑦 ( ·𝑠OLD ‘ 𝑈 ) 𝑥 ) ) = ( ( abs ‘ 𝑦 ) · ( 𝑁 ‘ 𝑥 ) ) ∧ ∀ 𝑦 ∈ 𝑋 ( 𝑁 ‘ ( 𝑥 ( +𝑣 ‘ 𝑈 ) 𝑦 ) ) ≤ ( ( 𝑁 ‘ 𝑥 ) + ( 𝑁 ‘ 𝑦 ) ) ) → ∀ 𝑥 ∈ 𝑋 ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ) |
10 |
|
fveqeq2 |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑁 ‘ 𝑥 ) = 0 ↔ ( 𝑁 ‘ 𝐴 ) = 0 ) ) |
11 |
|
eqeq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 = 𝑍 ↔ 𝐴 = 𝑍 ) ) |
12 |
10 11
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) ↔ ( ( 𝑁 ‘ 𝐴 ) = 0 → 𝐴 = 𝑍 ) ) ) |
13 |
12
|
rspccv |
⊢ ( ∀ 𝑥 ∈ 𝑋 ( ( 𝑁 ‘ 𝑥 ) = 0 → 𝑥 = 𝑍 ) → ( 𝐴 ∈ 𝑋 → ( ( 𝑁 ‘ 𝐴 ) = 0 → 𝐴 = 𝑍 ) ) ) |
14 |
7 9 13
|
3syl |
⊢ ( 𝑈 ∈ NrmCVec → ( 𝐴 ∈ 𝑋 → ( ( 𝑁 ‘ 𝐴 ) = 0 → 𝐴 = 𝑍 ) ) ) |
15 |
14
|
imp |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) = 0 → 𝐴 = 𝑍 ) ) |
16 |
|
fveq2 |
⊢ ( 𝐴 = 𝑍 → ( 𝑁 ‘ 𝐴 ) = ( 𝑁 ‘ 𝑍 ) ) |
17 |
2 3
|
nvz0 |
⊢ ( 𝑈 ∈ NrmCVec → ( 𝑁 ‘ 𝑍 ) = 0 ) |
18 |
16 17
|
sylan9eqr |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 = 𝑍 ) → ( 𝑁 ‘ 𝐴 ) = 0 ) |
19 |
18
|
ex |
⊢ ( 𝑈 ∈ NrmCVec → ( 𝐴 = 𝑍 → ( 𝑁 ‘ 𝐴 ) = 0 ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 = 𝑍 → ( 𝑁 ‘ 𝐴 ) = 0 ) ) |
21 |
15 20
|
impbid |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) = 0 ↔ 𝐴 = 𝑍 ) ) |