Step |
Hyp |
Ref |
Expression |
1 |
|
ressusp.1 |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
2 |
|
ressusp.2 |
⊢ 𝐽 = ( TopOpen ‘ 𝑊 ) |
3 |
|
ressuss |
⊢ ( 𝐴 ∈ 𝐽 → ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) = ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ) |
4 |
3
|
3ad2ant3 |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) = ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ) |
5 |
|
simp1 |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → 𝑊 ∈ UnifSp ) |
6 |
|
eqid |
⊢ ( UnifSt ‘ 𝑊 ) = ( UnifSt ‘ 𝑊 ) |
7 |
1 6 2
|
isusp |
⊢ ( 𝑊 ∈ UnifSp ↔ ( ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝐵 ) ∧ 𝐽 = ( unifTop ‘ ( UnifSt ‘ 𝑊 ) ) ) ) |
8 |
5 7
|
sylib |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝐵 ) ∧ 𝐽 = ( unifTop ‘ ( UnifSt ‘ 𝑊 ) ) ) ) |
9 |
8
|
simpld |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝐵 ) ) |
10 |
|
simp2 |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → 𝑊 ∈ TopSp ) |
11 |
1 2
|
istps |
⊢ ( 𝑊 ∈ TopSp ↔ 𝐽 ∈ ( TopOn ‘ 𝐵 ) ) |
12 |
10 11
|
sylib |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → 𝐽 ∈ ( TopOn ‘ 𝐵 ) ) |
13 |
|
simp3 |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ∈ 𝐽 ) |
14 |
|
toponss |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝐵 ) ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ⊆ 𝐵 ) |
15 |
12 13 14
|
syl2anc |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ⊆ 𝐵 ) |
16 |
|
trust |
⊢ ( ( ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝐵 ) ∧ 𝐴 ⊆ 𝐵 ) → ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ∈ ( UnifOn ‘ 𝐴 ) ) |
17 |
9 15 16
|
syl2anc |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ∈ ( UnifOn ‘ 𝐴 ) ) |
18 |
4 17
|
eqeltrd |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) ∈ ( UnifOn ‘ 𝐴 ) ) |
19 |
|
eqid |
⊢ ( 𝑊 ↾s 𝐴 ) = ( 𝑊 ↾s 𝐴 ) |
20 |
19 1
|
ressbas2 |
⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 = ( Base ‘ ( 𝑊 ↾s 𝐴 ) ) ) |
21 |
15 20
|
syl |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → 𝐴 = ( Base ‘ ( 𝑊 ↾s 𝐴 ) ) ) |
22 |
21
|
fveq2d |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( UnifOn ‘ 𝐴 ) = ( UnifOn ‘ ( Base ‘ ( 𝑊 ↾s 𝐴 ) ) ) ) |
23 |
18 22
|
eleqtrd |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) ∈ ( UnifOn ‘ ( Base ‘ ( 𝑊 ↾s 𝐴 ) ) ) ) |
24 |
8
|
simprd |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → 𝐽 = ( unifTop ‘ ( UnifSt ‘ 𝑊 ) ) ) |
25 |
13 24
|
eleqtrd |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → 𝐴 ∈ ( unifTop ‘ ( UnifSt ‘ 𝑊 ) ) ) |
26 |
|
restutopopn |
⊢ ( ( ( UnifSt ‘ 𝑊 ) ∈ ( UnifOn ‘ 𝐵 ) ∧ 𝐴 ∈ ( unifTop ‘ ( UnifSt ‘ 𝑊 ) ) ) → ( ( unifTop ‘ ( UnifSt ‘ 𝑊 ) ) ↾t 𝐴 ) = ( unifTop ‘ ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ) ) |
27 |
9 25 26
|
syl2anc |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( ( unifTop ‘ ( UnifSt ‘ 𝑊 ) ) ↾t 𝐴 ) = ( unifTop ‘ ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ) ) |
28 |
24
|
oveq1d |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( 𝐽 ↾t 𝐴 ) = ( ( unifTop ‘ ( UnifSt ‘ 𝑊 ) ) ↾t 𝐴 ) ) |
29 |
4
|
fveq2d |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( unifTop ‘ ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) ) = ( unifTop ‘ ( ( UnifSt ‘ 𝑊 ) ↾t ( 𝐴 × 𝐴 ) ) ) ) |
30 |
27 28 29
|
3eqtr4d |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( 𝐽 ↾t 𝐴 ) = ( unifTop ‘ ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) ) ) |
31 |
|
eqid |
⊢ ( Base ‘ ( 𝑊 ↾s 𝐴 ) ) = ( Base ‘ ( 𝑊 ↾s 𝐴 ) ) |
32 |
|
eqid |
⊢ ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) = ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) |
33 |
19 2
|
resstopn |
⊢ ( 𝐽 ↾t 𝐴 ) = ( TopOpen ‘ ( 𝑊 ↾s 𝐴 ) ) |
34 |
31 32 33
|
isusp |
⊢ ( ( 𝑊 ↾s 𝐴 ) ∈ UnifSp ↔ ( ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) ∈ ( UnifOn ‘ ( Base ‘ ( 𝑊 ↾s 𝐴 ) ) ) ∧ ( 𝐽 ↾t 𝐴 ) = ( unifTop ‘ ( UnifSt ‘ ( 𝑊 ↾s 𝐴 ) ) ) ) ) |
35 |
23 30 34
|
sylanbrc |
⊢ ( ( 𝑊 ∈ UnifSp ∧ 𝑊 ∈ TopSp ∧ 𝐴 ∈ 𝐽 ) → ( 𝑊 ↾s 𝐴 ) ∈ UnifSp ) |