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Theorem sxbrsiga 27889
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 27783 . . . 4  |- 𝔅  e.  (sigAlgebra `  RR )
2 eqid 2462 . . . . 5  |-  ran  (
e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )
32sxval 27789 . . . 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 672 . . 3  |-  (𝔅 ×s 𝔅 )  =  (sigaGen `  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) ) )
5 br2base 27868 . . . . 5  |-  U. ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  ( RR  X.  RR )
6 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
76tpr2uni 27511 . . . . 5  |-  U. ( J  tX  J )  =  ( RR  X.  RR )
85, 7eqtr4i 2494 . . . 4  |-  U. ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  U. ( J  tX  J )
9 brsigasspwrn 27784 . . . . . . . . . 10  |- 𝔅 
C_  ~P RR
109sseli 3495 . . . . . . . . 9  |-  ( e  e. 𝔅  ->  e  e.  ~P RR )
1110elpwid 4015 . . . . . . . 8  |-  ( e  e. 𝔅  ->  e  C_  RR )
129sseli 3495 . . . . . . . . 9  |-  ( f  e. 𝔅  ->  f  e.  ~P RR )
1312elpwid 4015 . . . . . . . 8  |-  ( f  e. 𝔅  ->  f  C_  RR )
14 xpinpreima2 27513 . . . . . . . 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 477 . . . . . . 7  |-  ( ( e  e. 𝔅  /\  f  e. 𝔅 )  ->  ( e  X.  f )  =  ( ( `' ( 1st  |`  ( RR  X.  RR ) ) " e
)  i^i  ( `' ( 2nd  |`  ( RR  X.  RR ) ) "
f ) ) )
166tpr2tp 27510 . . . . . . . . . 10  |-  ( J 
tX  J )  e.  (TopOn `  ( RR  X.  RR ) )
17 sigagensiga 27769 . . . . . . . . . 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 27754 . . . . . . . . 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 27770 . . . . . . . . . 10  |-  ( e  e. 𝔅  ->  (sigaGen `  ( J  tX  J ) )  e. 
U. ran sigAlgebra )
23 elrnsiga 27754 . . . . . . . . . . 11  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
241, 23mp1i 12 . . . . . . . . . 10  |-  ( e  e. 𝔅  -> 𝔅  e.  U.
ran sigAlgebra )
25 retopon 21000 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
266, 25eqeltri 2546 . . . . . . . . . . . . 13  |-  J  e.  (TopOn `  RR )
27 tx1cn 19840 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  RR )  /\  J  e.  (TopOn `  RR )
)  ->  ( 1st  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J
)  Cn  J ) )
2826, 26, 27mp2an 672 . . . . . . . . . . . 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 2463 . . . . . . . . . . 11  |-  ( e  e. 𝔅  ->  (sigaGen `  ( J  tX  J ) )  =  (sigaGen `  ( J  tX  J ) ) )
31 df-brsiga 27781 . . . . . . . . . . . . 13  |- 𝔅  =  (sigaGen `  ( topGen `
 ran  (,) )
)
326fveq2i 5862 . . . . . . . . . . . . 13  |-  (sigaGen `  J
)  =  (sigaGen `  ( topGen `
 ran  (,) )
)
3331, 32eqtr4i 2494 . . . . . . . . . . . 12  |- 𝔅  =  (sigaGen `  J
)
3433a1i 11 . . . . . . . . . . 11  |-  ( e  e. 𝔅  -> 𝔅  =  (sigaGen `  J ) )
3529, 30, 34cnmbfm 27862 . . . . . . . . . 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 27856 . . . . . . . . 9  |-  ( e  e. 𝔅  ->  ( `' ( 1st  |`  ( RR  X.  RR ) ) " e
)  e.  (sigaGen `  ( J  tX  J ) ) )
3837adantr 465 . . . . . . . 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 27770 . . . . . . . . . 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 19841 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  RR )  /\  J  e.  (TopOn `  RR )
)  ->  ( 2nd  |`  ( RR  X.  RR ) )  e.  ( ( J  tX  J
)  Cn  J ) )
4326, 26, 42mp2an 672 . . . . . . . . . . . 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 2463 . . . . . . . . . . 11  |-  ( f  e. 𝔅  ->  (sigaGen `  ( J  tX  J ) )  =  (sigaGen `  ( J  tX  J ) ) )
4633a1i 11 . . . . . . . . . . 11  |-  ( f  e. 𝔅  -> 𝔅  =  (sigaGen `  J ) )
4744, 45, 46cnmbfm 27862 . . . . . . . . . 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 27856 . . . . . . . . 9  |-  ( f  e. 𝔅  ->  ( `' ( 2nd  |`  ( RR  X.  RR ) ) " f
)  e.  (sigaGen `  ( J  tX  J ) ) )
5049adantl 466 . . . . . . . 8  |-  ( ( e  e. 𝔅  /\  f  e. 𝔅 )  ->  ( `' ( 2nd  |`  ( RR  X.  RR ) ) "
f )  e.  (sigaGen `  ( J  tX  J
) ) )
51 inelsiga 27763 . . . . . . . 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 1223 . . . . . . 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 2550 . . . . . 6  |-  ( ( e  e. 𝔅  /\  f  e. 𝔅 )  ->  ( e  X.  f )  e.  (sigaGen `  ( J  tX  J
) ) )
5453rgen2a 2886 . . . . 5  |-  A. e  e. 𝔅  A. f  e. 𝔅  ( e  X.  f
)  e.  (sigaGen `  ( J  tX  J ) )
55 eqid 2462 . . . . . 6  |-  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )
5655rnmpt2ss 27175 . . . . 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 27778 . . . 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 1319 . . 3  |-  (sigaGen `  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) ) )  C_  (sigaGen `  ( J  tX  J ) )
604, 59eqsstri 3529 . 2  |-  (𝔅 ×s 𝔅 ) 
C_  (sigaGen `  ( J  tX  J ) )
616sxbrsigalem6 27888 . 2  |-  (sigaGen `  ( J  tX  J ) ) 
C_  (𝔅 ×s 𝔅 )
6260, 61eqssi 3515 1  |-  (𝔅 ×s 𝔅 )  =  (sigaGen `  ( J  tX  J ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809    i^i cin 3470    C_ wss 3471   ~Pcpw 4005   U.cuni 4240    X. cxp 4992   `'ccnv 4993   ran crn 4995    |` cres 4996   "cima 4997   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6774   2ndc2nd 6775   RRcr 9482   (,)cioo 11520   topGenctg 14684  TopOnctopon 19157    Cn ccn 19486    tX ctx 19791  sigAlgebracsiga 27735  sigaGencsigagen 27766  𝔅cbrsiga 27780   ×s csx 27787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-ac2 8834  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-acn 8314  df-ac 8488  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460  df-ef 13656  df-sin 13658  df-cos 13659  df-pi 13661  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-refld 18403  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-cmp 19648  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-fcls 20172  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-cfil 21424  df-cmet 21426  df-cms 21504  df-limc 22000  df-dv 22001  df-log 22667  df-cxp 22668  df-logb 27635  df-siga 27736  df-sigagen 27767  df-brsiga 27781  df-sx 27788  df-mbfm 27850
This theorem is referenced by:  rrvadd  28019
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