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Theorem sxbrsiga 28417
Description: The product sigma-algebra  (𝔅 ×s 𝔅 ) 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  |-  (𝔅 ×s 𝔅 )  =  (sigaGen `  ( J  tX  J ) )

Proof of Theorem sxbrsiga
Dummy variables  e 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brsigarn 28311 . . . 4  |- 𝔅  e.  (sigAlgebra `  RR )
2 eqid 2382 . . . . 5  |-  ran  (
e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )
32sxval 28317 . . . 4  |-  ( (𝔅  e.  (sigAlgebra `
 RR )  /\ 𝔅  e.  (sigAlgebra `  RR ) )  -> 
(𝔅 ×s 𝔅 )  =  (sigaGen `  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) ) ) )
41, 1, 3mp2an 670 . . 3  |-  (𝔅 ×s 𝔅 )  =  (sigaGen `  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) ) )
5 br2base 28396 . . . . 5  |-  U. ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  ( RR  X.  RR )
6 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
76tpr2uni 28041 . . . . 5  |-  U. ( J  tX  J )  =  ( RR  X.  RR )
85, 7eqtr4i 2414 . . . 4  |-  U. ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  U. ( J  tX  J )
9 brsigasspwrn 28312 . . . . . . . . . 10  |- 𝔅 
C_  ~P RR
109sseli 3413 . . . . . . . . 9  |-  ( e  e. 𝔅  ->  e  e.  ~P RR )
1110elpwid 3937 . . . . . . . 8  |-  ( e  e. 𝔅  ->  e  C_  RR )
129sseli 3413 . . . . . . . . 9  |-  ( f  e. 𝔅  ->  f  e.  ~P RR )
1312elpwid 3937 . . . . . . . 8  |-  ( f  e. 𝔅  ->  f  C_  RR )
14 xpinpreima2 28043 . . . . . . . 8  |-  ( ( e  C_  RR  /\  f  C_  RR )  ->  (
e  X.  f )  =  ( ( `' ( 1st  |`  ( RR  X.  RR ) )
" e )  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) ) " f
) ) )
1511, 13, 14syl2an 475 . . . . . . 7  |-  ( ( e  e. 𝔅  /\  f  e. 𝔅 )  ->  ( e  X.  f )  =  ( ( `' ( 1st  |`  ( RR  X.  RR ) ) " e
)  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) ) "
f ) ) )
166tpr2tp 28040 . . . . . . . . . 10  |-  ( J 
tX  J )  e.  (TopOn `  ( RR  X.  RR ) )
17 sigagensiga 28290 . . . . . . . . . 10  |-  ( ( J  tX  J )  e.  (TopOn `  ( RR  X.  RR ) )  ->  (sigaGen `  ( J  tX  J ) )  e.  (sigAlgebra `  U. ( J 
tX  J ) ) )
1816, 17ax-mp 5 . . . . . . . . 9  |-  (sigaGen `  ( J  tX  J ) )  e.  (sigAlgebra `  U. ( J 
tX  J ) )
19 elrnsiga 28275 . . . . . . . . 9  |-  ( (sigaGen `  ( J  tX  J
) )  e.  (sigAlgebra ` 
U. ( J  tX  J ) )  -> 
(sigaGen `  ( J  tX  J ) )  e. 
U. ran sigAlgebra )
2018, 19mp1i 12 . . . . . . . 8  |-  ( ( e  e. 𝔅  /\  f  e. 𝔅 )  ->  (sigaGen `  ( J  tX  J ) )  e.  U. ran sigAlgebra )
2116a1i 11 . . . . . . . . . . 11  |-  ( e  e. 𝔅  ->  ( J  tX  J
)  e.  (TopOn `  ( RR  X.  RR ) ) )
2221sgsiga 28291 . . . . . . . . . 10  |-  ( e  e. 𝔅  ->  (sigaGen `  ( J  tX  J ) )  e. 
U. ran sigAlgebra )
23 elrnsiga 28275 . . . . . . . . . . 11  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
241, 23mp1i 12 . . . . . . . . . 10  |-  ( e  e. 𝔅  -> 𝔅  e.  U.
ran sigAlgebra )
25 retopon 21355 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
266, 25eqeltri 2466 . . . . . . . . . . . . 13  |-  J  e.  (TopOn `  RR )
27 tx1cn 20195 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  RR )  /\  J  e.  (TopOn `  RR )
)  ->  ( 1st  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J
)  Cn  J ) )
2826, 26, 27mp2an 670 . . . . . . . . . . . 12  |-  ( 1st  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J
)  Cn  J )
2928a1i 11 . . . . . . . . . . 11  |-  ( e  e. 𝔅  ->  ( 1st  |`  ( RR  X.  RR ) )  e.  ( ( J 
tX  J )  Cn  J ) )
30 eqidd 2383 . . . . . . . . . . 11  |-  ( e  e. 𝔅  ->  (sigaGen `  ( J  tX  J ) )  =  (sigaGen `  ( J  tX  J ) ) )
31 df-brsiga 28309 . . . . . . . . . . . . 13  |- 𝔅  =  (sigaGen `  ( topGen `
 ran  (,) )
)
326fveq2i 5777 . . . . . . . . . . . . 13  |-  (sigaGen `  J
)  =  (sigaGen `  ( topGen `
 ran  (,) )
)
3331, 32eqtr4i 2414 . . . . . . . . . . . 12  |- 𝔅  =  (sigaGen `  J
)
3433a1i 11 . . . . . . . . . . 11  |-  ( e  e. 𝔅  -> 𝔅  =  (sigaGen `  J ) )
3529, 30, 34cnmbfm 28390 . . . . . . . . . 10  |-  ( e  e. 𝔅  ->  ( 1st  |`  ( RR  X.  RR ) )  e.  ( (sigaGen `  ( J  tX  J ) )MblFnM𝔅 )
)
36 id 22 . . . . . . . . . 10  |-  ( e  e. 𝔅  ->  e  e. 𝔅 )
3722, 24, 35, 36mbfmcnvima 28384 . . . . . . . . 9  |-  ( e  e. 𝔅  ->  ( `' ( 1st  |`  ( RR  X.  RR ) ) " e
)  e.  (sigaGen `  ( J  tX  J ) ) )
3837adantr 463 . . . . . . . 8  |-  ( ( e  e. 𝔅  /\  f  e. 𝔅 )  ->  ( `' ( 1st  |`  ( RR  X.  RR ) ) "
e )  e.  (sigaGen `  ( J  tX  J
) ) )
3916a1i 11 . . . . . . . . . . 11  |-  ( f  e. 𝔅  ->  ( J  tX  J
)  e.  (TopOn `  ( RR  X.  RR ) ) )
4039sgsiga 28291 . . . . . . . . . 10  |-  ( f  e. 𝔅  ->  (sigaGen `  ( J  tX  J ) )  e. 
U. ran sigAlgebra )
411, 23mp1i 12 . . . . . . . . . 10  |-  ( f  e. 𝔅  -> 𝔅  e.  U.
ran sigAlgebra )
42 tx2cn 20196 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  RR )  /\  J  e.  (TopOn `  RR )
)  ->  ( 2nd  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J
)  Cn  J ) )
4326, 26, 42mp2an 670 . . . . . . . . . . . 12  |-  ( 2nd  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J
)  Cn  J )
4443a1i 11 . . . . . . . . . . 11  |-  ( f  e. 𝔅  ->  ( 2nd  |`  ( RR  X.  RR ) )  e.  ( ( J 
tX  J )  Cn  J ) )
45 eqidd 2383 . . . . . . . . . . 11  |-  ( f  e. 𝔅  ->  (sigaGen `  ( J  tX  J ) )  =  (sigaGen `  ( J  tX  J ) ) )
4633a1i 11 . . . . . . . . . . 11  |-  ( f  e. 𝔅  -> 𝔅  =  (sigaGen `  J ) )
4744, 45, 46cnmbfm 28390 . . . . . . . . . 10  |-  ( f  e. 𝔅  ->  ( 2nd  |`  ( RR  X.  RR ) )  e.  ( (sigaGen `  ( J  tX  J ) )MblFnM𝔅 )
)
48 id 22 . . . . . . . . . 10  |-  ( f  e. 𝔅  ->  f  e. 𝔅 )
4940, 41, 47, 48mbfmcnvima 28384 . . . . . . . . 9  |-  ( f  e. 𝔅  ->  ( `' ( 2nd  |`  ( RR  X.  RR ) ) " f
)  e.  (sigaGen `  ( J  tX  J ) ) )
5049adantl 464 . . . . . . . 8  |-  ( ( e  e. 𝔅  /\  f  e. 𝔅 )  ->  ( `' ( 2nd  |`  ( RR  X.  RR ) ) "
f )  e.  (sigaGen `  ( J  tX  J
) ) )
51 inelsiga 28284 . . . . . . . 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
) ) )
5220, 38, 50, 51syl3anc 1226 . . . . . . 7  |-  ( ( e  e. 𝔅  /\  f  e. 𝔅 )  ->  ( ( `' ( 1st  |`  ( RR  X.  RR ) )
" e )  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) ) " f
) )  e.  (sigaGen `  ( J  tX  J
) ) )
5315, 52eqeltrd 2470 . . . . . 6  |-  ( ( e  e. 𝔅  /\  f  e. 𝔅 )  ->  ( e  X.  f )  e.  (sigaGen `  ( J  tX  J
) ) )
5453rgen2a 2809 . . . . 5  |-  A. e  e. 𝔅  A. f  e. 𝔅  ( e  X.  f
)  e.  (sigaGen `  ( J  tX  J ) )
55 eqid 2382 . . . . . 6  |-  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )
5655rnmpt2ss 27661 . . . . 5  |-  ( A. e  e. 𝔅  A. f  e. 𝔅  ( e  X.  f
)  e.  (sigaGen `  ( J  tX  J ) )  ->  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  C_  (sigaGen `  ( J  tX  J
) ) )
5754, 56ax-mp 5 . . . 4  |-  ran  (
e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  C_  (sigaGen `  ( J  tX  J
) )
58 sigagenss2 28299 . . . 4  |-  ( ( U. ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  U. ( J  tX  J )  /\  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  C_  (sigaGen `  ( J  tX  J
) )  /\  ( J  tX  J )  e.  (TopOn `  ( RR  X.  RR ) ) )  ->  (sigaGen `  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) ) )  C_  (sigaGen `  ( J  tX  J ) ) )
598, 57, 16, 58mp3an 1322 . . 3  |-  (sigaGen `  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) ) )  C_  (sigaGen `  ( J  tX  J ) )
604, 59eqsstri 3447 . 2  |-  (𝔅 ×s 𝔅 ) 
C_  (sigaGen `  ( J  tX  J ) )
616sxbrsigalem6 28416 . 2  |-  (sigaGen `  ( J  tX  J ) ) 
C_  (𝔅 ×s 𝔅 )
6260, 61eqssi 3433 1  |-  (𝔅 ×s 𝔅 )  =  (sigaGen `  ( J  tX  J ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732    i^i cin 3388    C_ wss 3389   ~Pcpw 3927   U.cuni 4163    X. cxp 4911   `'ccnv 4912   ran crn 4914    |` cres 4915   "cima 4916   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   1stc1st 6697   2ndc2nd 6698   RRcr 9402   (,)cioo 11450   topGenctg 14845  TopOnctopon 19480    Cn ccn 19811    tX ctx 20146  sigAlgebracsiga 28256  sigaGencsigagen 28287  𝔅cbrsiga 28308   ×s csx 28315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-ac2 8756  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-omul 7053  df-er 7229  df-map 7340  df-pm 7341  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-acn 8236  df-ac 8410  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ioc 11455  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-mod 11897  df-seq 12011  df-exp 12070  df-fac 12256  df-bc 12283  df-hash 12308  df-shft 12902  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-limsup 13296  df-clim 13313  df-rlim 13314  df-sum 13511  df-ef 13805  df-sin 13807  df-cos 13808  df-pi 13810  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-fbas 18529  df-fg 18530  df-cnfld 18534  df-refld 18732  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-lp 19723  df-perf 19724  df-cn 19814  df-cnp 19815  df-haus 19902  df-cmp 19973  df-tx 20148  df-hmeo 20341  df-fil 20432  df-fm 20524  df-flim 20525  df-flf 20526  df-fcls 20527  df-xms 20908  df-ms 20909  df-tms 20910  df-cncf 21467  df-cfil 21779  df-cmet 21781  df-cms 21859  df-limc 22355  df-dv 22356  df-log 23029  df-cxp 23030  df-logb 23223  df-siga 28257  df-sigagen 28288  df-brsiga 28309  df-sx 28316  df-mbfm 28378
This theorem is referenced by:  rrvadd  28574
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