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Theorem dya2iocucvr 24587
Description: The dyadic rectangular set collection covers  ( RR  X.  RR ). (Contributed by Thierry Arnoux, 18-Sep-2017.)
Hypotheses
Ref Expression
sxbrsiga.0  |-  J  =  ( topGen `  ran  (,) )
dya2ioc.1  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
dya2ioc.2  |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I  |->  ( u  X.  v ) )
Assertion
Ref Expression
dya2iocucvr  |-  U. ran  R  =  ( RR  X.  RR )
Distinct variable groups:    x, n    x, I    v, u, I, x    u, n, v
Allowed substitution hints:    R( x, v, u, n)    I( n)    J( x, v, u, n)

Proof of Theorem dya2iocucvr
Dummy variables  c 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unissb 4005 . . 3  |-  ( U. ran  R  C_  ( RR  X.  RR )  <->  A. d  e.  ran  R  d  C_  ( RR  X.  RR ) )
2 dya2ioc.2 . . . . 5  |-  R  =  ( u  e.  ran  I ,  v  e.  ran  I  |->  ( u  X.  v ) )
3 vex 2919 . . . . . 6  |-  u  e. 
_V
4 vex 2919 . . . . . 6  |-  v  e. 
_V
53, 4xpex 4949 . . . . 5  |-  ( u  X.  v )  e. 
_V
62, 5elrnmpt2 6142 . . . 4  |-  ( d  e.  ran  R  <->  E. u  e.  ran  I E. v  e.  ran  I  d  =  ( u  X.  v
) )
7 simpr 448 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  =  ( u  X.  v ) )
8 pwssb 4137 . . . . . . . . . . . 12  |-  ( ran  I  C_  ~P RR  <->  A. d  e.  ran  I 
d  C_  RR )
9 dya2ioc.1 . . . . . . . . . . . . . 14  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
10 ovex 6065 . . . . . . . . . . . . . 14  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
119, 10elrnmpt2 6142 . . . . . . . . . . . . 13  |-  ( d  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
12 simpr 448 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  d  =  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
13 simpll 731 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  x  e.  ZZ )
1413zred 10331 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  x  e.  RR )
15 2re 10025 . . . . . . . . . . . . . . . . . . . 20  |-  2  e.  RR
1615a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  2  e.  RR )
17 2ne0 10039 . . . . . . . . . . . . . . . . . . . 20  |-  2  =/=  0
1817a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  2  =/=  0 )
19 simplr 732 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  n  e.  ZZ )
2016, 18, 19reexpclzd 11503 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( 2 ^ n )  e.  RR )
21 2cn 10026 . . . . . . . . . . . . . . . . . . . 20  |-  2  e.  CC
2221a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  2  e.  CC )
2322, 18, 19expne0d 11484 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( 2 ^ n )  =/=  0 )
2414, 20, 23redivcld 9798 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( x  /  ( 2 ^ n ) )  e.  RR )
25 1re 9046 . . . . . . . . . . . . . . . . . . . . 21  |-  1  e.  RR
2625a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  1  e.  RR )
2714, 26readdcld 9071 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( x  +  1 )  e.  RR )
2827, 20, 23redivcld 9798 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  +  1 )  /  ( 2 ^ n ) )  e.  RR )
2928rexrd 9090 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  +  1 )  /  ( 2 ^ n ) )  e. 
RR* )
30 icossre 10947 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  /  (
2 ^ n ) )  e.  RR  /\  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )  ->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) )  C_  RR )
3124, 29, 30syl2anc 643 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  C_  RR )
3212, 31eqsstrd 3342 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  d  C_  RR )
3332ex 424 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) )  ->  d  C_  RR ) )
3433rexlimivv 2795 . . . . . . . . . . . . 13  |-  ( E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  ->  d  C_  RR )
3511, 34sylbi 188 . . . . . . . . . . . 12  |-  ( d  e.  ran  I  -> 
d  C_  RR )
368, 35mprgbir 2736 . . . . . . . . . . 11  |-  ran  I  C_ 
~P RR
3736sseli 3304 . . . . . . . . . 10  |-  ( u  e.  ran  I  ->  u  e.  ~P RR )
3837elpwid 3768 . . . . . . . . 9  |-  ( u  e.  ran  I  ->  u  C_  RR )
3936sseli 3304 . . . . . . . . . 10  |-  ( v  e.  ran  I  -> 
v  e.  ~P RR )
4039elpwid 3768 . . . . . . . . 9  |-  ( v  e.  ran  I  -> 
v  C_  RR )
41 xpss12 4940 . . . . . . . . 9  |-  ( ( u  C_  RR  /\  v  C_  RR )  ->  (
u  X.  v ) 
C_  ( RR  X.  RR ) )
4238, 40, 41syl2an 464 . . . . . . . 8  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
u  X.  v ) 
C_  ( RR  X.  RR ) )
4342adantr 452 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
( u  X.  v
)  C_  ( RR  X.  RR ) )
447, 43eqsstrd 3342 . . . . . 6  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  C_  ( RR  X.  RR ) )
4544ex 424 . . . . 5  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
d  =  ( u  X.  v )  -> 
d  C_  ( RR  X.  RR ) ) )
4645rexlimivv 2795 . . . 4  |-  ( E. u  e.  ran  I E. v  e.  ran  I  d  =  (
u  X.  v )  ->  d  C_  ( RR  X.  RR ) )
476, 46sylbi 188 . . 3  |-  ( d  e.  ran  R  -> 
d  C_  ( RR  X.  RR ) )
481, 47mprgbir 2736 . 2  |-  U. ran  R 
C_  ( RR  X.  RR )
49 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
50 retop 18748 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
5149, 50eqeltri 2474 . . . . 5  |-  J  e. 
Top
5251, 51txtopi 17575 . . . 4  |-  ( J 
tX  J )  e. 
Top
53 uniretop 18749 . . . . . . 7  |-  RR  =  U. ( topGen `  ran  (,) )
5449unieqi 3985 . . . . . . 7  |-  U. J  =  U. ( topGen `  ran  (,) )
5553, 54eqtr4i 2427 . . . . . 6  |-  RR  =  U. J
5651, 51, 55, 55txunii 17578 . . . . 5  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
5756topopn 16934 . . . 4  |-  ( ( J  tX  J )  e.  Top  ->  ( RR  X.  RR )  e.  ( J  tX  J
) )
5849, 9, 2dya2iocuni 24586 . . . 4  |-  ( ( RR  X.  RR )  e.  ( J  tX  J )  ->  E. c  e.  ~P  ran  R U. c  =  ( RR  X.  RR ) )
5952, 57, 58mp2b 10 . . 3  |-  E. c  e.  ~P  ran  R U. c  =  ( RR  X.  RR )
60 simpr 448 . . . . 5  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  U. c  =  ( RR  X.  RR ) )
61 elpwi 3767 . . . . . . 7  |-  ( c  e.  ~P ran  R  ->  c  C_  ran  R )
6261adantr 452 . . . . . 6  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  c  C_ 
ran  R )
6362unissd 3999 . . . . 5  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  U. c  C_ 
U. ran  R )
6460, 63eqsstr3d 3343 . . . 4  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  ( RR  X.  RR )  C_  U.
ran  R )
6564rexlimiva 2785 . . 3  |-  ( E. c  e.  ~P  ran  R U. c  =  ( RR  X.  RR )  ->  ( RR  X.  RR )  C_  U. ran  R )
6659, 65ax-mp 8 . 2  |-  ( RR 
X.  RR )  C_  U.
ran  R
6748, 66eqssi 3324 1  |-  U. ran  R  =  ( RR  X.  RR )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667    C_ wss 3280   ~Pcpw 3759   U.cuni 3975    X. cxp 4835   ran crn 4838   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949   RR*cxr 9075    / cdiv 9633   2c2 10005   ZZcz 10238   (,)cioo 10872   [,)cico 10874   ^cexp 11337   topGenctg 13620   Topctop 16913    tX ctx 17545
This theorem is referenced by:  sxbrsigalem1  24588  sxbrsigalem2  24589  sxbrsigalem5  24591
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-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  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  ax-addf 9025  ax-mulf 9026
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-iin 4056  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-se 4502  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-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  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-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-logb 24342
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