Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sxbrsiga Structured version   Unicode version

Theorem sxbrsiga 28239
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 28133 . . . 4  |- 𝔅  e.  (sigAlgebra `  RR )
2 eqid 2443 . . . . 5  |-  ran  (
e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )
32sxval 28139 . . . 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 28218 . . . . 5  |-  U. ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  ( RR  X.  RR )
6 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
76tpr2uni 27865 . . . . 5  |-  U. ( J  tX  J )  =  ( RR  X.  RR )
85, 7eqtr4i 2475 . . . 4  |-  U. ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  U. ( J  tX  J )
9 brsigasspwrn 28134 . . . . . . . . . 10  |- 𝔅 
C_  ~P RR
109sseli 3485 . . . . . . . . 9  |-  ( e  e. 𝔅  ->  e  e.  ~P RR )
1110elpwid 4007 . . . . . . . 8  |-  ( e  e. 𝔅  ->  e  C_  RR )
129sseli 3485 . . . . . . . . 9  |-  ( f  e. 𝔅  ->  f  e.  ~P RR )
1312elpwid 4007 . . . . . . . 8  |-  ( f  e. 𝔅  ->  f  C_  RR )
14 xpinpreima2 27867 . . . . . . . 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 27864 . . . . . . . . . 10  |-  ( J 
tX  J )  e.  (TopOn `  ( RR  X.  RR ) )
17 sigagensiga 28119 . . . . . . . . . 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 28104 . . . . . . . . 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 28120 . . . . . . . . . 10  |-  ( e  e. 𝔅  ->  (sigaGen `  ( J  tX  J ) )  e. 
U. ran sigAlgebra )
23 elrnsiga 28104 . . . . . . . . . . 11  |-  (𝔅  e.  (sigAlgebra `  RR )  -> 𝔅  e.  U. ran sigAlgebra )
241, 23mp1i 12 . . . . . . . . . 10  |-  ( e  e. 𝔅  -> 𝔅  e.  U.
ran sigAlgebra )
25 retopon 21248 . . . . . . . . . . . . . 14  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
266, 25eqeltri 2527 . . . . . . . . . . . . 13  |-  J  e.  (TopOn `  RR )
27 tx1cn 20088 . . . . . . . . . . . . 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 2444 . . . . . . . . . . 11  |-  ( e  e. 𝔅  ->  (sigaGen `  ( J  tX  J ) )  =  (sigaGen `  ( J  tX  J ) ) )
31 df-brsiga 28131 . . . . . . . . . . . . 13  |- 𝔅  =  (sigaGen `  ( topGen `
 ran  (,) )
)
326fveq2i 5859 . . . . . . . . . . . . 13  |-  (sigaGen `  J
)  =  (sigaGen `  ( topGen `
 ran  (,) )
)
3331, 32eqtr4i 2475 . . . . . . . . . . . 12  |- 𝔅  =  (sigaGen `  J
)
3433a1i 11 . . . . . . . . . . 11  |-  ( e  e. 𝔅  -> 𝔅  =  (sigaGen `  J ) )
3529, 30, 34cnmbfm 28212 . . . . . . . . . 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 28206 . . . . . . . . 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 28120 . . . . . . . . . 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 20089 . . . . . . . . . . . . 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 2444 . . . . . . . . . . 11  |-  ( f  e. 𝔅  ->  (sigaGen `  ( J  tX  J ) )  =  (sigaGen `  ( J  tX  J ) ) )
4633a1i 11 . . . . . . . . . . 11  |-  ( f  e. 𝔅  -> 𝔅  =  (sigaGen `  J ) )
4744, 45, 46cnmbfm 28212 . . . . . . . . . 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 28206 . . . . . . . . 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 28113 . . . . . . . 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 1229 . . . . . . 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 2531 . . . . . 6  |-  ( ( e  e. 𝔅  /\  f  e. 𝔅 )  ->  ( e  X.  f )  e.  (sigaGen `  ( J  tX  J
) ) )
5453rgen2a 2870 . . . . 5  |-  A. e  e. 𝔅  A. f  e. 𝔅  ( e  X.  f
)  e.  (sigaGen `  ( J  tX  J ) )
55 eqid 2443 . . . . . 6  |-  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )  =  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) )
5655rnmpt2ss 27493 . . . . 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 28128 . . . 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 1325 . . 3  |-  (sigaGen `  ran  ( e  e. 𝔅 ,  f  e. 𝔅 
|->  ( e  X.  f
) ) )  C_  (sigaGen `  ( J  tX  J ) )
604, 59eqsstri 3519 . 2  |-  (𝔅 ×s 𝔅 ) 
C_  (sigaGen `  ( J  tX  J ) )
616sxbrsigalem6 28238 . 2  |-  (sigaGen `  ( J  tX  J ) ) 
C_  (𝔅 ×s 𝔅 )
6260, 61eqssi 3505 1  |-  (𝔅 ×s 𝔅 )  =  (sigaGen `  ( J  tX  J ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793    i^i cin 3460    C_ wss 3461   ~Pcpw 3997   U.cuni 4234    X. cxp 4987   `'ccnv 4988   ran crn 4990    |` cres 4991   "cima 4992   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   1stc1st 6783   2ndc2nd 6784   RRcr 9494   (,)cioo 11540   topGenctg 14817  TopOnctopon 19373    Cn ccn 19703    tX ctx 20039  sigAlgebracsiga 28085  sigaGencsigagen 28116  𝔅cbrsiga 28130   ×s csx 28137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-ac2 8846  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-ac 8500  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ioc 11545  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12882  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-limsup 13276  df-clim 13293  df-rlim 13294  df-sum 13491  df-ef 13785  df-sin 13787  df-cos 13788  df-pi 13790  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-mulg 16039  df-cntz 16334  df-cmn 16779  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-fbas 18395  df-fg 18396  df-cnfld 18400  df-refld 18619  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-lp 19615  df-perf 19616  df-cn 19706  df-cnp 19707  df-haus 19794  df-cmp 19865  df-tx 20041  df-hmeo 20234  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419  df-fcls 20420  df-xms 20801  df-ms 20802  df-tms 20803  df-cncf 21360  df-cfil 21672  df-cmet 21674  df-cms 21752  df-limc 22248  df-dv 22249  df-log 22922  df-cxp 22923  df-logb 27985  df-siga 28086  df-sigagen 28117  df-brsiga 28131  df-sx 28138  df-mbfm 28200
This theorem is referenced by:  rrvadd  28369
  Copyright terms: Public domain W3C validator