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Theorem sxbrsigalem3 24575
Description: The sigma-algebra generated by the closed half-spaces of  ( RR  X.  RR ) is a subset of the sigma-algebra generated by the closed sets of  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
Hypothesis
Ref Expression
sxbrsiga.0  |-  J  =  ( topGen `  ran  (,) )
Assertion
Ref Expression
sxbrsigalem3  |-  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) ) )  C_  (sigaGen `  ( Clsd `  ( J  tX  J ) ) )
Distinct variable group:    e, f
Allowed substitution hints:    J( e, f)

Proof of Theorem sxbrsigalem3
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sxbrsigalem0 24574 . . 3  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) )  =  ( RR  X.  RR )
2 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
3 retop 18748 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
42, 3eqeltri 2474 . . . . 5  |-  J  e. 
Top
54, 4txtopi 17575 . . . 4  |-  ( J 
tX  J )  e. 
Top
6 uniretop 18749 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
72unieqi 3985 . . . . . 6  |-  U. J  =  U. ( topGen `  ran  (,) )
86, 7eqtr4i 2427 . . . . 5  |-  RR  =  U. J
94, 4, 8, 8txunii 17578 . . . 4  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
105, 9unicls 24254 . . 3  |-  U. ( Clsd `  ( J  tX  J ) )  =  ( RR  X.  RR )
111, 10eqtr4i 2427 . 2  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) )  =  U. ( Clsd `  ( J  tX  J ) )
12 ovex 6065 . . . . . . 7  |-  ( e [,)  +oo )  e.  _V
13 reex 9037 . . . . . . 7  |-  RR  e.  _V
1412, 13xpex 4949 . . . . . 6  |-  ( ( e [,)  +oo )  X.  RR )  e.  _V
15 eqid 2404 . . . . . 6  |-  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )
1614, 15fnmpti 5532 . . . . 5  |-  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  Fn  RR
17 oveq1 6047 . . . . . . . . 9  |-  ( e  =  u  ->  (
e [,)  +oo )  =  ( u [,)  +oo ) )
1817xpeq1d 4860 . . . . . . . 8  |-  ( e  =  u  ->  (
( e [,)  +oo )  X.  RR )  =  ( ( u [,) 
+oo )  X.  RR ) )
19 ovex 6065 . . . . . . . . 9  |-  ( u [,)  +oo )  e.  _V
2019, 13xpex 4949 . . . . . . . 8  |-  ( ( u [,)  +oo )  X.  RR )  e.  _V
2118, 15, 20fvmpt 5765 . . . . . . 7  |-  ( u  e.  RR  ->  (
( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) ) `  u
)  =  ( ( u [,)  +oo )  X.  RR ) )
22 icopnfcld 18755 . . . . . . . . 9  |-  ( u  e.  RR  ->  (
u [,)  +oo )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
232fveq2i 5690 . . . . . . . . 9  |-  ( Clsd `  J )  =  (
Clsd `  ( topGen ` 
ran  (,) ) )
2422, 23syl6eleqr 2495 . . . . . . . 8  |-  ( u  e.  RR  ->  (
u [,)  +oo )  e.  ( Clsd `  J
) )
25 dif0 3658 . . . . . . . . 9  |-  ( RR 
\  (/) )  =  RR
26 0opn 16932 . . . . . . . . . . 11  |-  ( J  e.  Top  ->  (/)  e.  J
)
274, 26ax-mp 8 . . . . . . . . . 10  |-  (/)  e.  J
288opncld 17052 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  (/) 
e.  J )  -> 
( RR  \  (/) )  e.  ( Clsd `  J
) )
294, 27, 28mp2an 654 . . . . . . . . 9  |-  ( RR 
\  (/) )  e.  (
Clsd `  J )
3025, 29eqeltrri 2475 . . . . . . . 8  |-  RR  e.  ( Clsd `  J )
31 txcld 17588 . . . . . . . 8  |-  ( ( ( u [,)  +oo )  e.  ( Clsd `  J )  /\  RR  e.  ( Clsd `  J
) )  ->  (
( u [,)  +oo )  X.  RR )  e.  ( Clsd `  ( J  tX  J ) ) )
3224, 30, 31sylancl 644 . . . . . . 7  |-  ( u  e.  RR  ->  (
( u [,)  +oo )  X.  RR )  e.  ( Clsd `  ( J  tX  J ) ) )
3321, 32eqeltrd 2478 . . . . . 6  |-  ( u  e.  RR  ->  (
( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) ) `  u
)  e.  ( Clsd `  ( J  tX  J
) ) )
3433rgen 2731 . . . . 5  |-  A. u  e.  RR  ( ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) ) `  u )  e.  (
Clsd `  ( J  tX  J ) )
35 fnfvrnss 5855 . . . . 5  |-  ( ( ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  Fn  RR  /\ 
A. u  e.  RR  ( ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) ) `  u
)  e.  ( Clsd `  ( J  tX  J
) ) )  ->  ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  C_  ( Clsd `  ( J  tX  J ) ) )
3616, 34, 35mp2an 654 . . . 4  |-  ran  (
e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) ) 
C_  ( Clsd `  ( J  tX  J ) )
37 ovex 6065 . . . . . . 7  |-  ( f [,)  +oo )  e.  _V
3813, 37xpex 4949 . . . . . 6  |-  ( RR 
X.  ( f [,) 
+oo ) )  e. 
_V
39 eqid 2404 . . . . . 6  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) )
4038, 39fnmpti 5532 . . . . 5  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) )  Fn  RR
41 oveq1 6047 . . . . . . . . 9  |-  ( f  =  v  ->  (
f [,)  +oo )  =  ( v [,)  +oo ) )
4241xpeq2d 4861 . . . . . . . 8  |-  ( f  =  v  ->  ( RR  X.  ( f [,) 
+oo ) )  =  ( RR  X.  (
v [,)  +oo ) ) )
43 ovex 6065 . . . . . . . . 9  |-  ( v [,)  +oo )  e.  _V
4413, 43xpex 4949 . . . . . . . 8  |-  ( RR 
X.  ( v [,) 
+oo ) )  e. 
_V
4542, 39, 44fvmpt 5765 . . . . . . 7  |-  ( v  e.  RR  ->  (
( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) `  v )  =  ( RR  X.  ( v [,)  +oo ) ) )
46 icopnfcld 18755 . . . . . . . . 9  |-  ( v  e.  RR  ->  (
v [,)  +oo )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
4746, 23syl6eleqr 2495 . . . . . . . 8  |-  ( v  e.  RR  ->  (
v [,)  +oo )  e.  ( Clsd `  J
) )
48 txcld 17588 . . . . . . . 8  |-  ( ( RR  e.  ( Clsd `  J )  /\  (
v [,)  +oo )  e.  ( Clsd `  J
) )  ->  ( RR  X.  ( v [,) 
+oo ) )  e.  ( Clsd `  ( J  tX  J ) ) )
4930, 47, 48sylancr 645 . . . . . . 7  |-  ( v  e.  RR  ->  ( RR  X.  ( v [,) 
+oo ) )  e.  ( Clsd `  ( J  tX  J ) ) )
5045, 49eqeltrd 2478 . . . . . 6  |-  ( v  e.  RR  ->  (
( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) `  v )  e.  ( Clsd `  ( J  tX  J ) ) )
5150rgen 2731 . . . . 5  |-  A. v  e.  RR  ( ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) ) `
 v )  e.  ( Clsd `  ( J  tX  J ) )
52 fnfvrnss 5855 . . . . 5  |-  ( ( ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) )  Fn  RR  /\  A. v  e.  RR  (
( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) `  v )  e.  ( Clsd `  ( J  tX  J ) ) )  ->  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) ) 
C_  ( Clsd `  ( J  tX  J ) ) )
5340, 51, 52mp2an 654 . . . 4  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) ) 
C_  ( Clsd `  ( J  tX  J ) )
5436, 53unssi 3482 . . 3  |-  ( ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) )  C_  ( Clsd `  ( J  tX  J ) )
55 fvex 5701 . . . 4  |-  ( Clsd `  ( J  tX  J
) )  e.  _V
56 sssigagen 24481 . . . 4  |-  ( (
Clsd `  ( J  tX  J ) )  e. 
_V  ->  ( Clsd `  ( J  tX  J ) ) 
C_  (sigaGen `  ( Clsd `  ( J  tX  J
) ) ) )
5755, 56ax-mp 8 . . 3  |-  ( Clsd `  ( J  tX  J
) )  C_  (sigaGen `  ( Clsd `  ( J  tX  J ) ) )
5854, 57sstri 3317 . 2  |-  ( ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) )  C_  (sigaGen `  ( Clsd `  ( J  tX  J ) ) )
59 sigagenss2 24486 . 2  |-  ( ( U. ( ran  (
e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) ) )  =  U. ( Clsd `  ( J  tX  J ) )  /\  ( ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) )  C_  (sigaGen `  ( Clsd `  ( J  tX  J ) ) )  /\  ( Clsd `  ( J  tX  J
) )  e.  _V )  ->  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) ) )  C_  (sigaGen `  ( Clsd `  ( J  tX  J ) ) ) )
6011, 58, 55, 59mp3an 1279 1  |-  (sigaGen `  ( ran  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) ) )  C_  (sigaGen `  ( Clsd `  ( J  tX  J ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277    u. cun 3278    C_ wss 3280   (/)c0 3588   U.cuni 3975    e. cmpt 4226    X. cxp 4835   ran crn 4838    Fn wfn 5408   ` cfv 5413  (class class class)co 6040   RRcr 8945    +oocpnf 9073   (,)cioo 10872   [,)cico 10874   topGenctg 13620   Topctop 16913   Clsdccld 17035    tX ctx 17545  sigaGencsigagen 24474
This theorem is referenced by:  sxbrsigalem4  24590
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-ioo 10876  df-ico 10878  df-topgen 13622  df-top 16918  df-bases 16920  df-topon 16921  df-cld 17038  df-tx 17547  df-siga 24444  df-sigagen 24475
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