Theorem sxbrsiga 30352

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

Hypothesis

Ref	Expression
sxbrsiga.0	|- J = ( topGen ` ran (,) )

Assertion

Ref	Expression
sxbrsiga	|- (𝔅ℝ ×s 𝔅ℝ ) = (sigaGen ` ( J tX J ) )

Proof of Theorem sxbrsiga

Dummy variables e f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brsigarn 30247	. . . 4 |- 𝔅ℝ e. (sigAlgebra ` RR )
2		eqid 2622	. . . . 5 |- ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) = ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) )
3	2	sxval 30253	. . . 4 |- ( (𝔅ℝ e. (sigAlgebra ` RR ) /\ 𝔅ℝ e. (sigAlgebra ` RR ) ) -> (𝔅ℝ ×s 𝔅ℝ ) = (sigaGen ` ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) ) )
4	1, 1, 3	mp2an 708	. . 3 |- (𝔅ℝ ×s 𝔅ℝ ) = (sigaGen ` ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) )
5		br2base 30331	. . . . 5 |- U. ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) = ( RR X. RR )
6		sxbrsiga.0	. . . . . 6 |- J = ( topGen ` ran (,) )
7	6	tpr2uni 29951	. . . . 5 |- U. ( J tX J ) = ( RR X. RR )
8	5, 7	eqtr4i 2647	. . . 4 |- U. ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) = U. ( J tX J )
9		brsigasspwrn 30248	. . . . . . . . . 10 |- 𝔅ℝ C_ ~P RR
10	9	sseli 3599	. . . . . . . . 9 |- ( e e. 𝔅ℝ -> e e. ~P RR )
11	10	elpwid 4170	. . . . . . . 8 |- ( e e. 𝔅ℝ -> e C_ RR )
12	9	sseli 3599	. . . . . . . . 9 |- ( f e. 𝔅ℝ -> f e. ~P RR )
13	12	elpwid 4170	. . . . . . . 8 |- ( f e. 𝔅ℝ -> f C_ RR )
14		xpinpreima2 29953	. . . . . . . 8 |- ( ( e C_ RR /\ f C_ RR ) -> ( e X. f ) = ( ( `' ( 1st |` ( RR X. RR ) ) " e ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " f ) ) )
15	11, 13, 14	syl2an 494	. . . . . . 7 |- ( ( e e. 𝔅ℝ /\ f e. 𝔅ℝ ) -> ( e X. f ) = ( ( `' ( 1st |` ( RR X. RR ) ) " e ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " f ) ) )
16	6	tpr2tp 29950	. . . . . . . . . 10 |- ( J tX J ) e. (TopOn ` ( RR X. RR ) )
17		sigagensiga 30204	. . . . . . . . . 10 |- ( ( J tX J ) e. (TopOn ` ( RR X. RR ) ) -> (sigaGen ` ( J tX J ) ) e. (sigAlgebra ` U. ( J tX J ) ) )
18	16, 17	ax-mp 5	. . . . . . . . 9 |- (sigaGen ` ( J tX J ) ) e. (sigAlgebra ` U. ( J tX J ) )
19		elrnsiga 30189	. . . . . . . . 9 |- ( (sigaGen ` ( J tX J ) ) e. (sigAlgebra ` U. ( J tX J ) ) -> (sigaGen ` ( J tX J ) ) e. U. ran sigAlgebra )
20	18, 19	mp1i 13	. . . . . . . 8 |- ( ( e e. 𝔅ℝ /\ f e. 𝔅ℝ ) -> (sigaGen ` ( J tX J ) ) e. U. ran sigAlgebra )
21	16	a1i 11	. . . . . . . . . . 11 |- ( e e. 𝔅ℝ -> ( J tX J ) e. (TopOn ` ( RR X. RR ) ) )
22	21	sgsiga 30205	. . . . . . . . . 10 |- ( e e. 𝔅ℝ -> (sigaGen ` ( J tX J ) ) e. U. ran sigAlgebra )
23		elrnsiga 30189	. . . . . . . . . . 11 |- (𝔅ℝ e. (sigAlgebra ` RR ) -> 𝔅ℝ e. U. ran sigAlgebra )
24	1, 23	mp1i 13	. . . . . . . . . 10 |- ( e e. 𝔅ℝ -> 𝔅ℝ e. U. ran sigAlgebra )
25		retopon 22567	. . . . . . . . . . . . . 14 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
26	6, 25	eqeltri 2697	. . . . . . . . . . . . 13 |- J e. (TopOn ` RR )
27		tx1cn 21412	. . . . . . . . . . . . 13 |- ( ( J e. (TopOn ` RR ) /\ J e. (TopOn ` RR ) ) -> ( 1st |` ( RR X. RR ) ) e. ( ( J tX J ) Cn J ) )
28	26, 26, 27	mp2an 708	. . . . . . . . . . . 12 |- ( 1st |` ( RR X. RR ) ) e. ( ( J tX J ) Cn J )
29	28	a1i 11	. . . . . . . . . . 11 |- ( e e. 𝔅ℝ -> ( 1st |` ( RR X. RR ) ) e. ( ( J tX J ) Cn J ) )
30		eqidd 2623	. . . . . . . . . . 11 |- ( e e. 𝔅ℝ -> (sigaGen ` ( J tX J ) ) = (sigaGen ` ( J tX J ) ) )
31		df-brsiga 30245	. . . . . . . . . . . . 13 |- 𝔅ℝ = (sigaGen ` ( topGen ` ran (,) ) )
32	6	fveq2i 6194	. . . . . . . . . . . . 13 |- (sigaGen ` J ) = (sigaGen ` ( topGen ` ran (,) ) )
33	31, 32	eqtr4i 2647	. . . . . . . . . . . 12 |- 𝔅ℝ = (sigaGen ` J )
34	33	a1i 11	. . . . . . . . . . 11 |- ( e e. 𝔅ℝ -> 𝔅ℝ = (sigaGen ` J ) )
35	29, 30, 34	cnmbfm 30325	. . . . . . . . . 10 |- ( e e. 𝔅ℝ -> ( 1st |` ( RR X. RR ) ) e. ( (sigaGen ` ( J tX J ) )MblFnM𝔅ℝ ) )
36		id 22	. . . . . . . . . 10 |- ( e e. 𝔅ℝ -> e e. 𝔅ℝ )
37	22, 24, 35, 36	mbfmcnvima 30319	. . . . . . . . 9 |- ( e e. 𝔅ℝ -> ( `' ( 1st |` ( RR X. RR ) ) " e ) e. (sigaGen ` ( J tX J ) ) )
38	37	adantr 481	. . . . . . . 8 |- ( ( e e. 𝔅ℝ /\ f e. 𝔅ℝ ) -> ( `' ( 1st |` ( RR X. RR ) ) " e ) e. (sigaGen ` ( J tX J ) ) )
39	16	a1i 11	. . . . . . . . . . 11 |- ( f e. 𝔅ℝ -> ( J tX J ) e. (TopOn ` ( RR X. RR ) ) )
40	39	sgsiga 30205	. . . . . . . . . 10 |- ( f e. 𝔅ℝ -> (sigaGen ` ( J tX J ) ) e. U. ran sigAlgebra )
41	1, 23	mp1i 13	. . . . . . . . . 10 |- ( f e. 𝔅ℝ -> 𝔅ℝ e. U. ran sigAlgebra )
42		tx2cn 21413	. . . . . . . . . . . . 13 |- ( ( J e. (TopOn ` RR ) /\ J e. (TopOn ` RR ) ) -> ( 2nd |` ( RR X. RR ) ) e. ( ( J tX J ) Cn J ) )
43	26, 26, 42	mp2an 708	. . . . . . . . . . . 12 |- ( 2nd |` ( RR X. RR ) ) e. ( ( J tX J ) Cn J )
44	43	a1i 11	. . . . . . . . . . 11 |- ( f e. 𝔅ℝ -> ( 2nd |` ( RR X. RR ) ) e. ( ( J tX J ) Cn J ) )
45		eqidd 2623	. . . . . . . . . . 11 |- ( f e. 𝔅ℝ -> (sigaGen ` ( J tX J ) ) = (sigaGen ` ( J tX J ) ) )
46	33	a1i 11	. . . . . . . . . . 11 |- ( f e. 𝔅ℝ -> 𝔅ℝ = (sigaGen ` J ) )
47	44, 45, 46	cnmbfm 30325	. . . . . . . . . 10 |- ( f e. 𝔅ℝ -> ( 2nd |` ( RR X. RR ) ) e. ( (sigaGen ` ( J tX J ) )MblFnM𝔅ℝ ) )
48		id 22	. . . . . . . . . 10 |- ( f e. 𝔅ℝ -> f e. 𝔅ℝ )
49	40, 41, 47, 48	mbfmcnvima 30319	. . . . . . . . 9 |- ( f e. 𝔅ℝ -> ( `' ( 2nd |` ( RR X. RR ) ) " f ) e. (sigaGen ` ( J tX J ) ) )
50	49	adantl 482	. . . . . . . 8 |- ( ( e e. 𝔅ℝ /\ f e. 𝔅ℝ ) -> ( `' ( 2nd |` ( RR X. RR ) ) " f ) e. (sigaGen ` ( J tX J ) ) )
51		inelsiga 30198	. . . . . . . 8 |- ( ( (sigaGen ` ( J tX J ) ) e. U. ran sigAlgebra /\ ( `' ( 1st |` ( RR X. RR ) ) " e ) e. (sigaGen ` ( J tX J ) ) /\ ( `' ( 2nd |` ( RR X. RR ) ) " f ) e. (sigaGen ` ( J tX J ) ) ) -> ( ( `' ( 1st |` ( RR X. RR ) ) " e ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " f ) ) e. (sigaGen ` ( J tX J ) ) )
52	20, 38, 50, 51	syl3anc 1326	. . . . . . 7 |- ( ( e e. 𝔅ℝ /\ f e. 𝔅ℝ ) -> ( ( `' ( 1st |` ( RR X. RR ) ) " e ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " f ) ) e. (sigaGen ` ( J tX J ) ) )
53	15, 52	eqeltrd 2701	. . . . . 6 |- ( ( e e. 𝔅ℝ /\ f e. 𝔅ℝ ) -> ( e X. f ) e. (sigaGen ` ( J tX J ) ) )
54	53	rgen2a 2977	. . . . 5 |- A. e e. 𝔅ℝ A. f e. 𝔅ℝ ( e X. f ) e. (sigaGen ` ( J tX J ) )
55		eqid 2622	. . . . . 6 |- ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) = ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) )
56	55	rnmpt2ss 29473	. . . . 5 |- ( A. e e. 𝔅ℝ A. f e. 𝔅ℝ ( e X. f ) e. (sigaGen ` ( J tX J ) ) -> ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) C_ (sigaGen ` ( J tX J ) ) )
57	54, 56	ax-mp 5	. . . 4 |- ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) C_ (sigaGen ` ( J tX J ) )
58		sigagenss2 30213	. . . 4 |- ( ( U. ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) = U. ( J tX J ) /\ ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) C_ (sigaGen ` ( J tX J ) ) /\ ( J tX J ) e. (TopOn ` ( RR X. RR ) ) ) -> (sigaGen ` ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) ) C_ (sigaGen ` ( J tX J ) ) )
59	8, 57, 16, 58	mp3an 1424	. . 3 |- (sigaGen ` ran ( e e. 𝔅ℝ , f e. 𝔅ℝ |-> ( e X. f ) ) ) C_ (sigaGen ` ( J tX J ) )
60	4, 59	eqsstri 3635	. 2 |- (𝔅ℝ ×s 𝔅ℝ ) C_ (sigaGen ` ( J tX J ) )
61	6	sxbrsigalem6 30351	. 2 |- (sigaGen ` ( J tX J ) ) C_ (𝔅ℝ ×s 𝔅ℝ )
62	60, 61	eqssi 3619	1 |- (𝔅ℝ ×s 𝔅ℝ ) = (sigaGen ` ( J tX J ) )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   X. cxp 5112   `'ccnv 5113   ran crn 5115   |` cres 5116   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   RRcr 9935   (,)cioo 12175   topGenctg 16098  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363  sigAlgebracsiga 30170  sigaGencsigagen 30201  𝔅_ℝcbrsiga 30244   ×_s csx 30251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-refld 19951  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-fcls 21745  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-cfil 23053  df-cmet 23055  df-cms 23132  df-limc 23630  df-dv 23631  df-log 24303  df-cxp 24304  df-logb 24503  df-siga 30171  df-sigagen 30202  df-brsiga 30245  df-sx 30252  df-mbfm 30313

This theorem is referenced by:  rrvadd  30514