Step |
Hyp |
Ref |
Expression |
1 |
|
hhsst.1 |
⊢ 𝑈 = 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 |
2 |
|
hhsst.2 |
⊢ 𝑊 = 〈 〈 ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) , ( ·ℎ ↾ ( ℂ × 𝐻 ) ) 〉 , ( normℎ ↾ 𝐻 ) 〉 |
3 |
2
|
hhssnvt |
⊢ ( 𝐻 ∈ Sℋ → 𝑊 ∈ NrmCVec ) |
4 |
|
resss |
⊢ ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) ⊆ +ℎ |
5 |
|
resss |
⊢ ( ·ℎ ↾ ( ℂ × 𝐻 ) ) ⊆ ·ℎ |
6 |
|
resss |
⊢ ( normℎ ↾ 𝐻 ) ⊆ normℎ |
7 |
4 5 6
|
3pm3.2i |
⊢ ( ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) ⊆ +ℎ ∧ ( ·ℎ ↾ ( ℂ × 𝐻 ) ) ⊆ ·ℎ ∧ ( normℎ ↾ 𝐻 ) ⊆ normℎ ) |
8 |
3 7
|
jctir |
⊢ ( 𝐻 ∈ Sℋ → ( 𝑊 ∈ NrmCVec ∧ ( ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) ⊆ +ℎ ∧ ( ·ℎ ↾ ( ℂ × 𝐻 ) ) ⊆ ·ℎ ∧ ( normℎ ↾ 𝐻 ) ⊆ normℎ ) ) ) |
9 |
1
|
hhnv |
⊢ 𝑈 ∈ NrmCVec |
10 |
1
|
hhva |
⊢ +ℎ = ( +𝑣 ‘ 𝑈 ) |
11 |
2
|
hhssva |
⊢ ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) = ( +𝑣 ‘ 𝑊 ) |
12 |
1
|
hhsm |
⊢ ·ℎ = ( ·𝑠OLD ‘ 𝑈 ) |
13 |
2
|
hhsssm |
⊢ ( ·ℎ ↾ ( ℂ × 𝐻 ) ) = ( ·𝑠OLD ‘ 𝑊 ) |
14 |
1
|
hhnm |
⊢ normℎ = ( normCV ‘ 𝑈 ) |
15 |
2
|
hhssnm |
⊢ ( normℎ ↾ 𝐻 ) = ( normCV ‘ 𝑊 ) |
16 |
|
eqid |
⊢ ( SubSp ‘ 𝑈 ) = ( SubSp ‘ 𝑈 ) |
17 |
10 11 12 13 14 15 16
|
isssp |
⊢ ( 𝑈 ∈ NrmCVec → ( 𝑊 ∈ ( SubSp ‘ 𝑈 ) ↔ ( 𝑊 ∈ NrmCVec ∧ ( ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) ⊆ +ℎ ∧ ( ·ℎ ↾ ( ℂ × 𝐻 ) ) ⊆ ·ℎ ∧ ( normℎ ↾ 𝐻 ) ⊆ normℎ ) ) ) ) |
18 |
9 17
|
ax-mp |
⊢ ( 𝑊 ∈ ( SubSp ‘ 𝑈 ) ↔ ( 𝑊 ∈ NrmCVec ∧ ( ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) ⊆ +ℎ ∧ ( ·ℎ ↾ ( ℂ × 𝐻 ) ) ⊆ ·ℎ ∧ ( normℎ ↾ 𝐻 ) ⊆ normℎ ) ) ) |
19 |
8 18
|
sylibr |
⊢ ( 𝐻 ∈ Sℋ → 𝑊 ∈ ( SubSp ‘ 𝑈 ) ) |