Step |
Hyp |
Ref |
Expression |
1 |
|
sxbrsiga.0 |
⊢ 𝐽 = ( topGen ‘ ran (,) ) |
2 |
|
dya2ioc.1 |
⊢ 𝐼 = ( 𝑥 ∈ ℤ , 𝑛 ∈ ℤ ↦ ( ( 𝑥 / ( 2 ↑ 𝑛 ) ) [,) ( ( 𝑥 + 1 ) / ( 2 ↑ 𝑛 ) ) ) ) |
3 |
|
dya2ioc.2 |
⊢ 𝑅 = ( 𝑢 ∈ ran 𝐼 , 𝑣 ∈ ran 𝐼 ↦ ( 𝑢 × 𝑣 ) ) |
4 |
1 2 3
|
sxbrsigalem1 |
⊢ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ⊆ ( sigaGen ‘ ran 𝑅 ) |
5 |
1 2 3
|
sxbrsigalem2 |
⊢ ( sigaGen ‘ ran 𝑅 ) ⊆ ( sigaGen ‘ ( ran ( 𝑒 ∈ ℝ ↦ ( ( 𝑒 [,) +∞ ) × ℝ ) ) ∪ ran ( 𝑓 ∈ ℝ ↦ ( ℝ × ( 𝑓 [,) +∞ ) ) ) ) ) |
6 |
1
|
sxbrsigalem3 |
⊢ ( sigaGen ‘ ( ran ( 𝑒 ∈ ℝ ↦ ( ( 𝑒 [,) +∞ ) × ℝ ) ) ∪ ran ( 𝑓 ∈ ℝ ↦ ( ℝ × ( 𝑓 [,) +∞ ) ) ) ) ) ⊆ ( sigaGen ‘ ( Clsd ‘ ( 𝐽 ×t 𝐽 ) ) ) |
7 |
5 6
|
sstri |
⊢ ( sigaGen ‘ ran 𝑅 ) ⊆ ( sigaGen ‘ ( Clsd ‘ ( 𝐽 ×t 𝐽 ) ) ) |
8 |
1
|
tpr2tp |
⊢ ( 𝐽 ×t 𝐽 ) ∈ ( TopOn ‘ ( ℝ × ℝ ) ) |
9 |
8
|
topontopi |
⊢ ( 𝐽 ×t 𝐽 ) ∈ Top |
10 |
|
eqid |
⊢ ∪ ( 𝐽 ×t 𝐽 ) = ∪ ( 𝐽 ×t 𝐽 ) |
11 |
9 10
|
unicls |
⊢ ∪ ( Clsd ‘ ( 𝐽 ×t 𝐽 ) ) = ∪ ( 𝐽 ×t 𝐽 ) |
12 |
|
cldssbrsiga |
⊢ ( ( 𝐽 ×t 𝐽 ) ∈ Top → ( Clsd ‘ ( 𝐽 ×t 𝐽 ) ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
13 |
9 12
|
ax-mp |
⊢ ( Clsd ‘ ( 𝐽 ×t 𝐽 ) ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) |
14 |
|
sigagenss2 |
⊢ ( ( ∪ ( Clsd ‘ ( 𝐽 ×t 𝐽 ) ) = ∪ ( 𝐽 ×t 𝐽 ) ∧ ( Clsd ‘ ( 𝐽 ×t 𝐽 ) ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∧ ( 𝐽 ×t 𝐽 ) ∈ Top ) → ( sigaGen ‘ ( Clsd ‘ ( 𝐽 ×t 𝐽 ) ) ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
15 |
11 13 9 14
|
mp3an |
⊢ ( sigaGen ‘ ( Clsd ‘ ( 𝐽 ×t 𝐽 ) ) ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) |
16 |
7 15
|
sstri |
⊢ ( sigaGen ‘ ran 𝑅 ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) |
17 |
4 16
|
eqssi |
⊢ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) = ( sigaGen ‘ ran 𝑅 ) |