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Theorem dya2iocucvr 24626
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 4037 . . 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 2951 . . . . . 6  |-  u  e. 
_V
4 vex 2951 . . . . . 6  |-  v  e. 
_V
53, 4xpex 4982 . . . . 5  |-  ( u  X.  v )  e. 
_V
62, 5elrnmpt2 6175 . . . 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 4169 . . . . . . . . . . . 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 6098 . . . . . . . . . . . . . 14  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
119, 10elrnmpt2 6175 . . . . . . . . . . . . 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 10367 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  x  e.  RR )
15 2re 10061 . . . . . . . . . . . . . . . . . . . 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 10075 . . . . . . . . . . . . . . . . . . . 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 11540 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( 2 ^ n )  e.  RR )
21 2cn 10062 . . . . . . . . . . . . . . . . . . . 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 11521 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( 2 ^ n )  =/=  0 )
2414, 20, 23redivcld 9834 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( x  /  ( 2 ^ n ) )  e.  RR )
25 1re 9082 . . . . . . . . . . . . . . . . . . . . 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 9107 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( x  +  1 )  e.  RR )
2827, 20, 23redivcld 9834 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  +  1 )  /  ( 2 ^ n ) )  e.  RR )
2928rexrd 9126 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  +  1 )  /  ( 2 ^ n ) )  e. 
RR* )
30 icossre 10983 . . . . . . . . . . . . . . . . 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 3374 . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . 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 2768 . . . . . . . . . . 11  |-  ran  I  C_ 
~P RR
3736sseli 3336 . . . . . . . . . 10  |-  ( u  e.  ran  I  ->  u  e.  ~P RR )
3837elpwid 3800 . . . . . . . . 9  |-  ( u  e.  ran  I  ->  u  C_  RR )
3936sseli 3336 . . . . . . . . . 10  |-  ( v  e.  ran  I  -> 
v  e.  ~P RR )
4039elpwid 3800 . . . . . . . . 9  |-  ( v  e.  ran  I  -> 
v  C_  RR )
41 xpss12 4973 . . . . . . . . 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 3374 . . . . . 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 2827 . . . 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 2768 . 2  |-  U. ran  R 
C_  ( RR  X.  RR )
49 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
50 retop 18787 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
5149, 50eqeltri 2505 . . . . 5  |-  J  e. 
Top
5251, 51txtopi 17614 . . . 4  |-  ( J 
tX  J )  e. 
Top
53 uniretop 18788 . . . . . . 7  |-  RR  =  U. ( topGen `  ran  (,) )
5449unieqi 4017 . . . . . . 7  |-  U. J  =  U. ( topGen `  ran  (,) )
5553, 54eqtr4i 2458 . . . . . 6  |-  RR  =  U. J
5651, 51, 55, 55txunii 17617 . . . . 5  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
5756topopn 16971 . . . 4  |-  ( ( J  tX  J )  e.  Top  ->  ( RR  X.  RR )  e.  ( J  tX  J
) )
5849, 9, 2dya2iocuni 24625 . . . 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 3799 . . . . . . 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 4031 . . . . 5  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  U. c  C_ 
U. ran  R )
6460, 63eqsstr3d 3375 . . . 4  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  ( RR  X.  RR )  C_  U.
ran  R )
6564rexlimiva 2817 . . 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 3356 1  |-  U. ran  R  =  ( RR  X.  RR )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    C_ wss 3312   ~Pcpw 3791   U.cuni 4007    X. cxp 4868   ran crn 4871   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985   RR*cxr 9111    / cdiv 9669   2c2 10041   ZZcz 10274   (,)cioo 10908   [,)cico 10910   ^cexp 11374   topGenctg 13657   Topctop 16950    tX ctx 17584
This theorem is referenced by:  sxbrsigalem1  24627  sxbrsigalem2  24628  sxbrsigalem5  24630
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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  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  ax-addf 9061  ax-mulf 9062
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-iin 4088  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-se 4534  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-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  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-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-pi 12667  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446  df-cxp 20447  df-logb 24381
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