Step |
Hyp |
Ref |
Expression |
1 |
|
ncvsprp.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
ncvsprp.n |
⊢ 𝑁 = ( norm ‘ 𝑊 ) |
3 |
|
ncvsprp.s |
⊢ · = ( ·𝑠 ‘ 𝑊 ) |
4 |
|
simpl |
⊢ ( ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ∧ 𝐴 ∈ 𝑉 ) → 𝑊 ∈ ( NrmVec ∩ ℂVec ) ) |
5 |
|
elin |
⊢ ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ↔ ( 𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec ) ) |
6 |
|
id |
⊢ ( 𝑊 ∈ ℂVec → 𝑊 ∈ ℂVec ) |
7 |
6
|
cvsclm |
⊢ ( 𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod ) |
8 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
9 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
10 |
8 9
|
clmneg1 |
⊢ ( 𝑊 ∈ ℂMod → - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
11 |
7 10
|
syl |
⊢ ( 𝑊 ∈ ℂVec → - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
12 |
5 11
|
simplbiim |
⊢ ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) → - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
13 |
12
|
adantr |
⊢ ( ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ∧ 𝐴 ∈ 𝑉 ) → - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) |
14 |
|
simpr |
⊢ ( ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ∧ 𝐴 ∈ 𝑉 ) → 𝐴 ∈ 𝑉 ) |
15 |
1 2 3 8 9
|
ncvsprp |
⊢ ( ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ∧ - 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐴 ∈ 𝑉 ) → ( 𝑁 ‘ ( - 1 · 𝐴 ) ) = ( ( abs ‘ - 1 ) · ( 𝑁 ‘ 𝐴 ) ) ) |
16 |
4 13 14 15
|
syl3anc |
⊢ ( ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ∧ 𝐴 ∈ 𝑉 ) → ( 𝑁 ‘ ( - 1 · 𝐴 ) ) = ( ( abs ‘ - 1 ) · ( 𝑁 ‘ 𝐴 ) ) ) |
17 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
18 |
17
|
absnegi |
⊢ ( abs ‘ - 1 ) = ( abs ‘ 1 ) |
19 |
|
abs1 |
⊢ ( abs ‘ 1 ) = 1 |
20 |
18 19
|
eqtri |
⊢ ( abs ‘ - 1 ) = 1 |
21 |
20
|
oveq1i |
⊢ ( ( abs ‘ - 1 ) · ( 𝑁 ‘ 𝐴 ) ) = ( 1 · ( 𝑁 ‘ 𝐴 ) ) |
22 |
|
nvcnlm |
⊢ ( 𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod ) |
23 |
|
nlmngp |
⊢ ( 𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp ) |
24 |
22 23
|
syl |
⊢ ( 𝑊 ∈ NrmVec → 𝑊 ∈ NrmGrp ) |
25 |
24
|
adantr |
⊢ ( ( 𝑊 ∈ NrmVec ∧ 𝑊 ∈ ℂVec ) → 𝑊 ∈ NrmGrp ) |
26 |
5 25
|
sylbi |
⊢ ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) → 𝑊 ∈ NrmGrp ) |
27 |
1 2
|
nmcl |
⊢ ( ( 𝑊 ∈ NrmGrp ∧ 𝐴 ∈ 𝑉 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℝ ) |
28 |
26 27
|
sylan |
⊢ ( ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ∧ 𝐴 ∈ 𝑉 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℝ ) |
29 |
28
|
recnd |
⊢ ( ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ∧ 𝐴 ∈ 𝑉 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℂ ) |
30 |
29
|
mulid2d |
⊢ ( ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ∧ 𝐴 ∈ 𝑉 ) → ( 1 · ( 𝑁 ‘ 𝐴 ) ) = ( 𝑁 ‘ 𝐴 ) ) |
31 |
21 30
|
syl5eq |
⊢ ( ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ∧ 𝐴 ∈ 𝑉 ) → ( ( abs ‘ - 1 ) · ( 𝑁 ‘ 𝐴 ) ) = ( 𝑁 ‘ 𝐴 ) ) |
32 |
16 31
|
eqtrd |
⊢ ( ( 𝑊 ∈ ( NrmVec ∩ ℂVec ) ∧ 𝐴 ∈ 𝑉 ) → ( 𝑁 ‘ ( - 1 · 𝐴 ) ) = ( 𝑁 ‘ 𝐴 ) ) |