Step |
Hyp |
Ref |
Expression |
1 |
|
hhss.1 |
⊢ 𝑊 = 〈 〈 ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) , ( ·ℎ ↾ ( ℂ × 𝐻 ) ) 〉 , ( normℎ ↾ 𝐻 ) 〉 |
2 |
|
eqid |
⊢ ( ·𝑠OLD ‘ 𝑊 ) = ( ·𝑠OLD ‘ 𝑊 ) |
3 |
2
|
smfval |
⊢ ( ·𝑠OLD ‘ 𝑊 ) = ( 2nd ‘ ( 1st ‘ 𝑊 ) ) |
4 |
1
|
fveq2i |
⊢ ( 1st ‘ 𝑊 ) = ( 1st ‘ 〈 〈 ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) , ( ·ℎ ↾ ( ℂ × 𝐻 ) ) 〉 , ( normℎ ↾ 𝐻 ) 〉 ) |
5 |
|
opex |
⊢ 〈 ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) , ( ·ℎ ↾ ( ℂ × 𝐻 ) ) 〉 ∈ V |
6 |
|
normf |
⊢ normℎ : ℋ ⟶ ℝ |
7 |
|
ax-hilex |
⊢ ℋ ∈ V |
8 |
|
fex |
⊢ ( ( normℎ : ℋ ⟶ ℝ ∧ ℋ ∈ V ) → normℎ ∈ V ) |
9 |
6 7 8
|
mp2an |
⊢ normℎ ∈ V |
10 |
9
|
resex |
⊢ ( normℎ ↾ 𝐻 ) ∈ V |
11 |
5 10
|
op1st |
⊢ ( 1st ‘ 〈 〈 ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) , ( ·ℎ ↾ ( ℂ × 𝐻 ) ) 〉 , ( normℎ ↾ 𝐻 ) 〉 ) = 〈 ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) , ( ·ℎ ↾ ( ℂ × 𝐻 ) ) 〉 |
12 |
4 11
|
eqtri |
⊢ ( 1st ‘ 𝑊 ) = 〈 ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) , ( ·ℎ ↾ ( ℂ × 𝐻 ) ) 〉 |
13 |
12
|
fveq2i |
⊢ ( 2nd ‘ ( 1st ‘ 𝑊 ) ) = ( 2nd ‘ 〈 ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) , ( ·ℎ ↾ ( ℂ × 𝐻 ) ) 〉 ) |
14 |
|
hilablo |
⊢ +ℎ ∈ AbelOp |
15 |
|
resexg |
⊢ ( +ℎ ∈ AbelOp → ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) ∈ V ) |
16 |
14 15
|
ax-mp |
⊢ ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) ∈ V |
17 |
|
hvmulex |
⊢ ·ℎ ∈ V |
18 |
17
|
resex |
⊢ ( ·ℎ ↾ ( ℂ × 𝐻 ) ) ∈ V |
19 |
16 18
|
op2nd |
⊢ ( 2nd ‘ 〈 ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) , ( ·ℎ ↾ ( ℂ × 𝐻 ) ) 〉 ) = ( ·ℎ ↾ ( ℂ × 𝐻 ) ) |
20 |
3 13 19
|
3eqtrri |
⊢ ( ·ℎ ↾ ( ℂ × 𝐻 ) ) = ( ·𝑠OLD ‘ 𝑊 ) |