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Theorem dya2iocucvr 29058
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 4193 . . 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 3025 . . . . . 6  |-  u  e. 
_V
4 vex 3025 . . . . . 6  |-  v  e. 
_V
53, 4xpex 6553 . . . . 5  |-  ( u  X.  v )  e. 
_V
62, 5elrnmpt2 6367 . . . 4  |-  ( d  e.  ran  R  <->  E. u  e.  ran  I E. v  e.  ran  I  d  =  ( u  X.  v
) )
7 simpr 462 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  =  ( u  X.  v ) )
8 pwssb 4332 . . . . . . . . . . . 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 6277 . . . . . . . . . . . . . 14  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
119, 10elrnmpt2 6367 . . . . . . . . . . . . 13  |-  ( d  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
12 simpr 462 . . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  x  e.  ZZ )
1413zred 10991 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  x  e.  RR )
15 2re 10630 . . . . . . . . . . . . . . . . . . . 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 10653 . . . . . . . . . . . . . . . . . . . 20  |-  2  =/=  0
1817a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  2  =/=  0 )
19 simplr 760 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  n  e.  ZZ )
2016, 18, 19reexpclzd 12391 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( 2 ^ n )  e.  RR )
21 2cnd 10633 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  2  e.  CC )
2221, 18, 19expne0d 12372 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( 2 ^ n )  =/=  0 )
2314, 20, 22redivcld 10386 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( x  /  ( 2 ^ n ) )  e.  RR )
24 1red 9609 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  1  e.  RR )
2514, 24readdcld 9621 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( x  +  1 )  e.  RR )
2625, 20, 22redivcld 10386 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  +  1 )  /  ( 2 ^ n ) )  e.  RR )
2726rexrd 9641 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  +  1 )  /  ( 2 ^ n ) )  e. 
RR* )
28 icossre 11666 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  /  (
2 ^ n ) )  e.  RR  /\  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )  ->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) )  C_  RR )
2923, 27, 28syl2anc 665 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  C_  RR )
3012, 29eqsstrd 3441 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  d  C_  RR )
3130ex 435 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) )  ->  d  C_  RR ) )
3231rexlimivv 2861 . . . . . . . . . . . . 13  |-  ( E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  ->  d  C_  RR )
3311, 32sylbi 198 . . . . . . . . . . . 12  |-  ( d  e.  ran  I  -> 
d  C_  RR )
348, 33mprgbir 2729 . . . . . . . . . . 11  |-  ran  I  C_ 
~P RR
3534sseli 3403 . . . . . . . . . 10  |-  ( u  e.  ran  I  ->  u  e.  ~P RR )
3635elpwid 3934 . . . . . . . . 9  |-  ( u  e.  ran  I  ->  u  C_  RR )
3734sseli 3403 . . . . . . . . . 10  |-  ( v  e.  ran  I  -> 
v  e.  ~P RR )
3837elpwid 3934 . . . . . . . . 9  |-  ( v  e.  ran  I  -> 
v  C_  RR )
39 xpss12 4902 . . . . . . . . 9  |-  ( ( u  C_  RR  /\  v  C_  RR )  ->  (
u  X.  v ) 
C_  ( RR  X.  RR ) )
4036, 38, 39syl2an 479 . . . . . . . 8  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
u  X.  v ) 
C_  ( RR  X.  RR ) )
4140adantr 466 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
( u  X.  v
)  C_  ( RR  X.  RR ) )
427, 41eqsstrd 3441 . . . . . 6  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  C_  ( RR  X.  RR ) )
4342ex 435 . . . . 5  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
d  =  ( u  X.  v )  -> 
d  C_  ( RR  X.  RR ) ) )
4443rexlimivv 2861 . . . 4  |-  ( E. u  e.  ran  I E. v  e.  ran  I  d  =  (
u  X.  v )  ->  d  C_  ( RR  X.  RR ) )
456, 44sylbi 198 . . 3  |-  ( d  e.  ran  R  -> 
d  C_  ( RR  X.  RR ) )
461, 45mprgbir 2729 . 2  |-  U. ran  R 
C_  ( RR  X.  RR )
47 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
48 retop 21724 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
4947, 48eqeltri 2502 . . . . 5  |-  J  e. 
Top
5049, 49txtopi 20547 . . . 4  |-  ( J 
tX  J )  e. 
Top
51 uniretop 21725 . . . . . . 7  |-  RR  =  U. ( topGen `  ran  (,) )
5247unieqi 4171 . . . . . . 7  |-  U. J  =  U. ( topGen `  ran  (,) )
5351, 52eqtr4i 2453 . . . . . 6  |-  RR  =  U. J
5449, 49, 53, 53txunii 20550 . . . . 5  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
5554topopn 19878 . . . 4  |-  ( ( J  tX  J )  e.  Top  ->  ( RR  X.  RR )  e.  ( J  tX  J
) )
5647, 9, 2dya2iocuni 29057 . . . 4  |-  ( ( RR  X.  RR )  e.  ( J  tX  J )  ->  E. c  e.  ~P  ran  R U. c  =  ( RR  X.  RR ) )
5750, 55, 56mp2b 10 . . 3  |-  E. c  e.  ~P  ran  R U. c  =  ( RR  X.  RR )
58 simpr 462 . . . . 5  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  U. c  =  ( RR  X.  RR ) )
59 elpwi 3933 . . . . . . 7  |-  ( c  e.  ~P ran  R  ->  c  C_  ran  R )
6059adantr 466 . . . . . 6  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  c  C_ 
ran  R )
6160unissd 4186 . . . . 5  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  U. c  C_ 
U. ran  R )
6258, 61eqsstr3d 3442 . . . 4  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  ( RR  X.  RR )  C_  U.
ran  R )
6362rexlimiva 2852 . . 3  |-  ( E. c  e.  ~P  ran  R U. c  =  ( RR  X.  RR )  ->  ( RR  X.  RR )  C_  U. ran  R )
6457, 63ax-mp 5 . 2  |-  ( RR 
X.  RR )  C_  U.
ran  R
6546, 64eqssi 3423 1  |-  U. ran  R  =  ( RR  X.  RR )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   E.wrex 2715    C_ wss 3379   ~Pcpw 3924   U.cuni 4162    X. cxp 4794   ran crn 4797   ` cfv 5544  (class class class)co 6249    |-> cmpt2 6251   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493   RR*cxr 9625    / cdiv 10220   2c2 10610   ZZcz 10888   (,)cioo 11586   [,)cico 11588   ^cexp 12222   topGenctg 15279   Topctop 19859    tX ctx 20517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-inf2 8099  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-iin 4245  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-se 4756  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-isom 5553  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-of 6489  df-om 6651  df-1st 6751  df-2nd 6752  df-supp 6870  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-2o 7138  df-oadd 7141  df-er 7318  df-map 7429  df-pm 7430  df-ixp 7478  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-fsupp 7837  df-fi 7878  df-sup 7909  df-inf 7910  df-oi 7978  df-card 8325  df-cda 8549  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-9 10626  df-10 10627  df-n0 10821  df-z 10889  df-dec 11003  df-uz 11111  df-q 11216  df-rp 11254  df-xneg 11360  df-xadd 11361  df-xmul 11362  df-ioo 11590  df-ioc 11591  df-ico 11592  df-icc 11593  df-fz 11736  df-fzo 11867  df-fl 11978  df-mod 12047  df-seq 12164  df-exp 12223  df-fac 12410  df-bc 12438  df-hash 12466  df-shft 13074  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-limsup 13469  df-clim 13495  df-rlim 13496  df-sum 13696  df-ef 14064  df-sin 14066  df-cos 14067  df-pi 14069  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-mulr 15147  df-starv 15148  df-sca 15149  df-vsca 15150  df-ip 15151  df-tset 15152  df-ple 15153  df-ds 15155  df-unif 15156  df-hom 15157  df-cco 15158  df-rest 15264  df-topn 15265  df-0g 15283  df-gsum 15284  df-topgen 15285  df-pt 15286  df-prds 15289  df-xrs 15343  df-qtop 15349  df-imas 15350  df-xps 15353  df-mre 15435  df-mrc 15436  df-acs 15438  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-submnd 16526  df-mulg 16619  df-cntz 16914  df-cmn 17375  df-psmet 18905  df-xmet 18906  df-met 18907  df-bl 18908  df-mopn 18909  df-fbas 18910  df-fg 18911  df-cnfld 18914  df-refld 19115  df-top 19863  df-bases 19864  df-topon 19865  df-topsp 19866  df-cld 19976  df-ntr 19977  df-cls 19978  df-nei 20056  df-lp 20094  df-perf 20095  df-cn 20185  df-cnp 20186  df-haus 20273  df-cmp 20344  df-tx 20519  df-hmeo 20712  df-fil 20803  df-fm 20895  df-flim 20896  df-flf 20897  df-fcls 20898  df-xms 21277  df-ms 21278  df-tms 21279  df-cncf 21852  df-cfil 22167  df-cmet 22169  df-cms 22245  df-limc 22763  df-dv 22764  df-log 23448  df-cxp 23449  df-logb 23644
This theorem is referenced by:  sxbrsigalem1  29059  sxbrsigalem2  29060  sxbrsigalem5  29062
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