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Theorem sxbrsigalem3 24614
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 24613 . . 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 18787 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
42, 3eqeltri 2505 . . . . 5  |-  J  e. 
Top
54, 4txtopi 17614 . . . 4  |-  ( J 
tX  J )  e. 
Top
6 uniretop 18788 . . . . . 6  |-  RR  =  U. ( topGen `  ran  (,) )
72unieqi 4017 . . . . . 6  |-  U. J  =  U. ( topGen `  ran  (,) )
86, 7eqtr4i 2458 . . . . 5  |-  RR  =  U. J
94, 4, 8, 8txunii 17617 . . . 4  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
105, 9unicls 24293 . . 3  |-  U. ( Clsd `  ( J  tX  J ) )  =  ( RR  X.  RR )
111, 10eqtr4i 2458 . 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 6098 . . . . . . 7  |-  ( e [,)  +oo )  e.  _V
13 reex 9073 . . . . . . 7  |-  RR  e.  _V
1412, 13xpex 4982 . . . . . 6  |-  ( ( e [,)  +oo )  X.  RR )  e.  _V
15 eqid 2435 . . . . . 6  |-  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )
1614, 15fnmpti 5565 . . . . 5  |-  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  Fn  RR
17 oveq1 6080 . . . . . . . . 9  |-  ( e  =  u  ->  (
e [,)  +oo )  =  ( u [,)  +oo ) )
1817xpeq1d 4893 . . . . . . . 8  |-  ( e  =  u  ->  (
( e [,)  +oo )  X.  RR )  =  ( ( u [,) 
+oo )  X.  RR ) )
19 ovex 6098 . . . . . . . . 9  |-  ( u [,)  +oo )  e.  _V
2019, 13xpex 4982 . . . . . . . 8  |-  ( ( u [,)  +oo )  X.  RR )  e.  _V
2118, 15, 20fvmpt 5798 . . . . . . 7  |-  ( u  e.  RR  ->  (
( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) ) `  u
)  =  ( ( u [,)  +oo )  X.  RR ) )
22 icopnfcld 18794 . . . . . . . . 9  |-  ( u  e.  RR  ->  (
u [,)  +oo )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
232fveq2i 5723 . . . . . . . . 9  |-  ( Clsd `  J )  =  (
Clsd `  ( topGen ` 
ran  (,) ) )
2422, 23syl6eleqr 2526 . . . . . . . 8  |-  ( u  e.  RR  ->  (
u [,)  +oo )  e.  ( Clsd `  J
) )
25 dif0 3690 . . . . . . . . 9  |-  ( RR 
\  (/) )  =  RR
26 0opn 16969 . . . . . . . . . . 11  |-  ( J  e.  Top  ->  (/)  e.  J
)
274, 26ax-mp 8 . . . . . . . . . 10  |-  (/)  e.  J
288opncld 17089 . . . . . . . . . 10  |-  ( ( J  e.  Top  /\  (/) 
e.  J )  -> 
( RR  \  (/) )  e.  ( Clsd `  J
) )
294, 27, 28mp2an 654 . . . . . . . . 9  |-  ( RR 
\  (/) )  e.  (
Clsd `  J )
3025, 29eqeltrri 2506 . . . . . . . 8  |-  RR  e.  ( Clsd `  J )
31 txcld 17627 . . . . . . . 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 2509 . . . . . 6  |-  ( u  e.  RR  ->  (
( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) ) `  u
)  e.  ( Clsd `  ( J  tX  J
) ) )
3433rgen 2763 . . . . 5  |-  A. u  e.  RR  ( ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) ) `  u )  e.  (
Clsd `  ( J  tX  J ) )
35 fnfvrnss 5888 . . . . 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 6098 . . . . . . 7  |-  ( f [,)  +oo )  e.  _V
3813, 37xpex 4982 . . . . . 6  |-  ( RR 
X.  ( f [,) 
+oo ) )  e. 
_V
39 eqid 2435 . . . . . 6  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) )
4038, 39fnmpti 5565 . . . . 5  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) )  Fn  RR
41 oveq1 6080 . . . . . . . . 9  |-  ( f  =  v  ->  (
f [,)  +oo )  =  ( v [,)  +oo ) )
4241xpeq2d 4894 . . . . . . . 8  |-  ( f  =  v  ->  ( RR  X.  ( f [,) 
+oo ) )  =  ( RR  X.  (
v [,)  +oo ) ) )
43 ovex 6098 . . . . . . . . 9  |-  ( v [,)  +oo )  e.  _V
4413, 43xpex 4982 . . . . . . . 8  |-  ( RR 
X.  ( v [,) 
+oo ) )  e. 
_V
4542, 39, 44fvmpt 5798 . . . . . . 7  |-  ( v  e.  RR  ->  (
( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) `  v )  =  ( RR  X.  ( v [,)  +oo ) ) )
46 icopnfcld 18794 . . . . . . . . 9  |-  ( v  e.  RR  ->  (
v [,)  +oo )  e.  ( Clsd `  ( topGen `
 ran  (,) )
) )
4746, 23syl6eleqr 2526 . . . . . . . 8  |-  ( v  e.  RR  ->  (
v [,)  +oo )  e.  ( Clsd `  J
) )
48 txcld 17627 . . . . . . . 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 2509 . . . . . 6  |-  ( v  e.  RR  ->  (
( f  e.  RR  |->  ( RR  X.  (
f [,)  +oo ) ) ) `  v )  e.  ( Clsd `  ( J  tX  J ) ) )
5150rgen 2763 . . . . 5  |-  A. v  e.  RR  ( ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) ) `
 v )  e.  ( Clsd `  ( J  tX  J ) )
52 fnfvrnss 5888 . . . . 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 3514 . . 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 5734 . . . 4  |-  ( Clsd `  ( J  tX  J
) )  e.  _V
56 sssigagen 24520 . . . 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 3349 . 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 24525 . 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   (/)c0 3620   U.cuni 4007    e. cmpt 4258    X. cxp 4868   ran crn 4871    Fn wfn 5441   ` cfv 5446  (class class class)co 6073   RRcr 8981    +oocpnf 9109   (,)cioo 10908   [,)cico 10910   topGenctg 13657   Topctop 16950   Clsdccld 17072    tX ctx 17584  sigaGencsigagen 24513
This theorem is referenced by:  sxbrsigalem4  24629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-ioo 10912  df-ico 10914  df-topgen 13659  df-top 16955  df-bases 16957  df-topon 16958  df-cld 17075  df-tx 17586  df-siga 24483  df-sigagen 24514
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