Step |
Hyp |
Ref |
Expression |
1 |
|
sspz.z |
⊢ 𝑍 = ( 0vec ‘ 𝑈 ) |
2 |
|
sspz.q |
⊢ 𝑄 = ( 0vec ‘ 𝑊 ) |
3 |
|
sspz.h |
⊢ 𝐻 = ( SubSp ‘ 𝑈 ) |
4 |
3
|
sspnv |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ NrmCVec ) |
5 |
|
eqid |
⊢ ( BaseSet ‘ 𝑊 ) = ( BaseSet ‘ 𝑊 ) |
6 |
5 2
|
nvzcl |
⊢ ( 𝑊 ∈ NrmCVec → 𝑄 ∈ ( BaseSet ‘ 𝑊 ) ) |
7 |
6 6
|
jca |
⊢ ( 𝑊 ∈ NrmCVec → ( 𝑄 ∈ ( BaseSet ‘ 𝑊 ) ∧ 𝑄 ∈ ( BaseSet ‘ 𝑊 ) ) ) |
8 |
4 7
|
syl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑄 ∈ ( BaseSet ‘ 𝑊 ) ∧ 𝑄 ∈ ( BaseSet ‘ 𝑊 ) ) ) |
9 |
|
eqid |
⊢ ( −𝑣 ‘ 𝑈 ) = ( −𝑣 ‘ 𝑈 ) |
10 |
|
eqid |
⊢ ( −𝑣 ‘ 𝑊 ) = ( −𝑣 ‘ 𝑊 ) |
11 |
5 9 10 3
|
sspmval |
⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ ( BaseSet ‘ 𝑊 ) ∧ 𝑄 ∈ ( BaseSet ‘ 𝑊 ) ) ) → ( 𝑄 ( −𝑣 ‘ 𝑊 ) 𝑄 ) = ( 𝑄 ( −𝑣 ‘ 𝑈 ) 𝑄 ) ) |
12 |
8 11
|
mpdan |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑄 ( −𝑣 ‘ 𝑊 ) 𝑄 ) = ( 𝑄 ( −𝑣 ‘ 𝑈 ) 𝑄 ) ) |
13 |
5 10 2
|
nvmid |
⊢ ( ( 𝑊 ∈ NrmCVec ∧ 𝑄 ∈ ( BaseSet ‘ 𝑊 ) ) → ( 𝑄 ( −𝑣 ‘ 𝑊 ) 𝑄 ) = 𝑄 ) |
14 |
4 6 13
|
syl2anc2 |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑄 ( −𝑣 ‘ 𝑊 ) 𝑄 ) = 𝑄 ) |
15 |
|
eqid |
⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 𝑈 ) |
16 |
15 5 3
|
sspba |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( BaseSet ‘ 𝑊 ) ⊆ ( BaseSet ‘ 𝑈 ) ) |
17 |
4 6
|
syl |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑄 ∈ ( BaseSet ‘ 𝑊 ) ) |
18 |
16 17
|
sseldd |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑄 ∈ ( BaseSet ‘ 𝑈 ) ) |
19 |
15 9 1
|
nvmid |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑄 ∈ ( BaseSet ‘ 𝑈 ) ) → ( 𝑄 ( −𝑣 ‘ 𝑈 ) 𝑄 ) = 𝑍 ) |
20 |
18 19
|
syldan |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑄 ( −𝑣 ‘ 𝑈 ) 𝑄 ) = 𝑍 ) |
21 |
12 14 20
|
3eqtr3d |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑄 = 𝑍 ) |