Metamath Proof Explorer


Theorem sxbrsiga

Description: The product sigma-algebra ( BrSiga sX BrSiga ) is the Borel algebra on ( RR X. RR ) See example 5.1.1 of Cohn p. 143 . (Contributed by Thierry Arnoux, 10-Oct-2017)

Ref Expression
Hypothesis sxbrsiga.0 𝐽 = ( topGen ‘ ran (,) )
Assertion sxbrsiga ( 𝔅 ×s 𝔅 ) = ( sigaGen ‘ ( 𝐽 ×t 𝐽 ) )

Proof

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