Step |
Hyp |
Ref |
Expression |
1 |
|
sxbrsiga.0 |
⊢ 𝐽 = ( topGen ‘ ran (,) ) |
2 |
|
brsigarn |
⊢ 𝔅ℝ ∈ ( sigAlgebra ‘ ℝ ) |
3 |
|
eqid |
⊢ ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) = ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) |
4 |
3
|
sxval |
⊢ ( ( 𝔅ℝ ∈ ( sigAlgebra ‘ ℝ ) ∧ 𝔅ℝ ∈ ( sigAlgebra ‘ ℝ ) ) → ( 𝔅ℝ ×s 𝔅ℝ ) = ( sigaGen ‘ ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) ) ) |
5 |
2 2 4
|
mp2an |
⊢ ( 𝔅ℝ ×s 𝔅ℝ ) = ( sigaGen ‘ ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) ) |
6 |
|
br2base |
⊢ ∪ ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) = ( ℝ × ℝ ) |
7 |
1
|
tpr2uni |
⊢ ∪ ( 𝐽 ×t 𝐽 ) = ( ℝ × ℝ ) |
8 |
6 7
|
eqtr4i |
⊢ ∪ ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) = ∪ ( 𝐽 ×t 𝐽 ) |
9 |
|
brsigasspwrn |
⊢ 𝔅ℝ ⊆ 𝒫 ℝ |
10 |
9
|
sseli |
⊢ ( 𝑒 ∈ 𝔅ℝ → 𝑒 ∈ 𝒫 ℝ ) |
11 |
10
|
elpwid |
⊢ ( 𝑒 ∈ 𝔅ℝ → 𝑒 ⊆ ℝ ) |
12 |
9
|
sseli |
⊢ ( 𝑓 ∈ 𝔅ℝ → 𝑓 ∈ 𝒫 ℝ ) |
13 |
12
|
elpwid |
⊢ ( 𝑓 ∈ 𝔅ℝ → 𝑓 ⊆ ℝ ) |
14 |
|
xpinpreima2 |
⊢ ( ( 𝑒 ⊆ ℝ ∧ 𝑓 ⊆ ℝ ) → ( 𝑒 × 𝑓 ) = ( ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ 𝑒 ) ∩ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ 𝑓 ) ) ) |
15 |
11 13 14
|
syl2an |
⊢ ( ( 𝑒 ∈ 𝔅ℝ ∧ 𝑓 ∈ 𝔅ℝ ) → ( 𝑒 × 𝑓 ) = ( ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ 𝑒 ) ∩ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ 𝑓 ) ) ) |
16 |
1
|
tpr2tp |
⊢ ( 𝐽 ×t 𝐽 ) ∈ ( TopOn ‘ ( ℝ × ℝ ) ) |
17 |
|
sigagensiga |
⊢ ( ( 𝐽 ×t 𝐽 ) ∈ ( TopOn ‘ ( ℝ × ℝ ) ) → ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∈ ( sigAlgebra ‘ ∪ ( 𝐽 ×t 𝐽 ) ) ) |
18 |
16 17
|
ax-mp |
⊢ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∈ ( sigAlgebra ‘ ∪ ( 𝐽 ×t 𝐽 ) ) |
19 |
|
elrnsiga |
⊢ ( ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∈ ( sigAlgebra ‘ ∪ ( 𝐽 ×t 𝐽 ) ) → ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∈ ∪ ran sigAlgebra ) |
20 |
18 19
|
mp1i |
⊢ ( ( 𝑒 ∈ 𝔅ℝ ∧ 𝑓 ∈ 𝔅ℝ ) → ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∈ ∪ ran sigAlgebra ) |
21 |
16
|
a1i |
⊢ ( 𝑒 ∈ 𝔅ℝ → ( 𝐽 ×t 𝐽 ) ∈ ( TopOn ‘ ( ℝ × ℝ ) ) ) |
22 |
21
|
sgsiga |
⊢ ( 𝑒 ∈ 𝔅ℝ → ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∈ ∪ ran sigAlgebra ) |
23 |
|
elrnsiga |
⊢ ( 𝔅ℝ ∈ ( sigAlgebra ‘ ℝ ) → 𝔅ℝ ∈ ∪ ran sigAlgebra ) |
24 |
2 23
|
mp1i |
⊢ ( 𝑒 ∈ 𝔅ℝ → 𝔅ℝ ∈ ∪ ran sigAlgebra ) |
25 |
|
retopon |
⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) |
26 |
1 25
|
eqeltri |
⊢ 𝐽 ∈ ( TopOn ‘ ℝ ) |
27 |
|
tx1cn |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ 𝐽 ∈ ( TopOn ‘ ℝ ) ) → ( 1st ↾ ( ℝ × ℝ ) ) ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) |
28 |
26 26 27
|
mp2an |
⊢ ( 1st ↾ ( ℝ × ℝ ) ) ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) |
29 |
28
|
a1i |
⊢ ( 𝑒 ∈ 𝔅ℝ → ( 1st ↾ ( ℝ × ℝ ) ) ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) |
30 |
|
eqidd |
⊢ ( 𝑒 ∈ 𝔅ℝ → ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) = ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
31 |
|
df-brsiga |
⊢ 𝔅ℝ = ( sigaGen ‘ ( topGen ‘ ran (,) ) ) |
32 |
1
|
fveq2i |
⊢ ( sigaGen ‘ 𝐽 ) = ( sigaGen ‘ ( topGen ‘ ran (,) ) ) |
33 |
31 32
|
eqtr4i |
⊢ 𝔅ℝ = ( sigaGen ‘ 𝐽 ) |
34 |
33
|
a1i |
⊢ ( 𝑒 ∈ 𝔅ℝ → 𝔅ℝ = ( sigaGen ‘ 𝐽 ) ) |
35 |
29 30 34
|
cnmbfm |
⊢ ( 𝑒 ∈ 𝔅ℝ → ( 1st ↾ ( ℝ × ℝ ) ) ∈ ( ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) MblFnM 𝔅ℝ ) ) |
36 |
|
id |
⊢ ( 𝑒 ∈ 𝔅ℝ → 𝑒 ∈ 𝔅ℝ ) |
37 |
22 24 35 36
|
mbfmcnvima |
⊢ ( 𝑒 ∈ 𝔅ℝ → ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ 𝑒 ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝑒 ∈ 𝔅ℝ ∧ 𝑓 ∈ 𝔅ℝ ) → ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ 𝑒 ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
39 |
16
|
a1i |
⊢ ( 𝑓 ∈ 𝔅ℝ → ( 𝐽 ×t 𝐽 ) ∈ ( TopOn ‘ ( ℝ × ℝ ) ) ) |
40 |
39
|
sgsiga |
⊢ ( 𝑓 ∈ 𝔅ℝ → ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∈ ∪ ran sigAlgebra ) |
41 |
2 23
|
mp1i |
⊢ ( 𝑓 ∈ 𝔅ℝ → 𝔅ℝ ∈ ∪ ran sigAlgebra ) |
42 |
|
tx2cn |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ 𝐽 ∈ ( TopOn ‘ ℝ ) ) → ( 2nd ↾ ( ℝ × ℝ ) ) ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) |
43 |
26 26 42
|
mp2an |
⊢ ( 2nd ↾ ( ℝ × ℝ ) ) ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) |
44 |
43
|
a1i |
⊢ ( 𝑓 ∈ 𝔅ℝ → ( 2nd ↾ ( ℝ × ℝ ) ) ∈ ( ( 𝐽 ×t 𝐽 ) Cn 𝐽 ) ) |
45 |
|
eqidd |
⊢ ( 𝑓 ∈ 𝔅ℝ → ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) = ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
46 |
33
|
a1i |
⊢ ( 𝑓 ∈ 𝔅ℝ → 𝔅ℝ = ( sigaGen ‘ 𝐽 ) ) |
47 |
44 45 46
|
cnmbfm |
⊢ ( 𝑓 ∈ 𝔅ℝ → ( 2nd ↾ ( ℝ × ℝ ) ) ∈ ( ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) MblFnM 𝔅ℝ ) ) |
48 |
|
id |
⊢ ( 𝑓 ∈ 𝔅ℝ → 𝑓 ∈ 𝔅ℝ ) |
49 |
40 41 47 48
|
mbfmcnvima |
⊢ ( 𝑓 ∈ 𝔅ℝ → ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ 𝑓 ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
50 |
49
|
adantl |
⊢ ( ( 𝑒 ∈ 𝔅ℝ ∧ 𝑓 ∈ 𝔅ℝ ) → ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ 𝑓 ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
51 |
|
inelsiga |
⊢ ( ( ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∈ ∪ ran sigAlgebra ∧ ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ 𝑒 ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∧ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ 𝑓 ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) → ( ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ 𝑒 ) ∩ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ 𝑓 ) ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
52 |
20 38 50 51
|
syl3anc |
⊢ ( ( 𝑒 ∈ 𝔅ℝ ∧ 𝑓 ∈ 𝔅ℝ ) → ( ( ◡ ( 1st ↾ ( ℝ × ℝ ) ) “ 𝑒 ) ∩ ( ◡ ( 2nd ↾ ( ℝ × ℝ ) ) “ 𝑓 ) ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
53 |
15 52
|
eqeltrd |
⊢ ( ( 𝑒 ∈ 𝔅ℝ ∧ 𝑓 ∈ 𝔅ℝ ) → ( 𝑒 × 𝑓 ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
54 |
53
|
rgen2 |
⊢ ∀ 𝑒 ∈ 𝔅ℝ ∀ 𝑓 ∈ 𝔅ℝ ( 𝑒 × 𝑓 ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) |
55 |
|
eqid |
⊢ ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) = ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) |
56 |
55
|
rnmposs |
⊢ ( ∀ 𝑒 ∈ 𝔅ℝ ∀ 𝑓 ∈ 𝔅ℝ ( 𝑒 × 𝑓 ) ∈ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) → ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
57 |
54 56
|
ax-mp |
⊢ ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) |
58 |
|
sigagenss2 |
⊢ ( ( ∪ ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) = ∪ ( 𝐽 ×t 𝐽 ) ∧ ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ∧ ( 𝐽 ×t 𝐽 ) ∈ ( TopOn ‘ ( ℝ × ℝ ) ) ) → ( sigaGen ‘ ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ) |
59 |
8 57 16 58
|
mp3an |
⊢ ( sigaGen ‘ ran ( 𝑒 ∈ 𝔅ℝ , 𝑓 ∈ 𝔅ℝ ↦ ( 𝑒 × 𝑓 ) ) ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) |
60 |
5 59
|
eqsstri |
⊢ ( 𝔅ℝ ×s 𝔅ℝ ) ⊆ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) |
61 |
1
|
sxbrsigalem6 |
⊢ ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) ⊆ ( 𝔅ℝ ×s 𝔅ℝ ) |
62 |
60 61
|
eqssi |
⊢ ( 𝔅ℝ ×s 𝔅ℝ ) = ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) ) |