Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dya2iocucvr Structured version   Unicode version

Theorem dya2iocucvr 27923
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 4277 . . 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 3116 . . . . . 6  |-  u  e. 
_V
4 vex 3116 . . . . . 6  |-  v  e. 
_V
53, 4xpex 6588 . . . . 5  |-  ( u  X.  v )  e. 
_V
62, 5elrnmpt2 6399 . . . 4  |-  ( d  e.  ran  R  <->  E. u  e.  ran  I E. v  e.  ran  I  d  =  ( u  X.  v
) )
7 simpr 461 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  =  ( u  X.  v ) )
8 pwssb 4412 . . . . . . . . . . . 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 6309 . . . . . . . . . . . . . 14  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
119, 10elrnmpt2 6399 . . . . . . . . . . . . 13  |-  ( d  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
12 simpr 461 . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  x  e.  ZZ )
1413zred 10966 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  x  e.  RR )
15 2re 10605 . . . . . . . . . . . . . . . . . . . 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 10628 . . . . . . . . . . . . . . . . . . . 20  |-  2  =/=  0
1817a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  2  =/=  0 )
19 simplr 754 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  n  e.  ZZ )
2016, 18, 19reexpclzd 12303 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( 2 ^ n )  e.  RR )
21 2cnd 10608 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  2  e.  CC )
2221, 18, 19expne0d 12284 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( 2 ^ n )  =/=  0 )
2314, 20, 22redivcld 10372 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( x  /  ( 2 ^ n ) )  e.  RR )
24 1re 9595 . . . . . . . . . . . . . . . . . . . . 21  |-  1  e.  RR
2524a1i 11 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  1  e.  RR )
2614, 25readdcld 9623 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( x  +  1 )  e.  RR )
2726, 20, 22redivcld 10372 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  +  1 )  /  ( 2 ^ n ) )  e.  RR )
2827rexrd 9643 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  +  1 )  /  ( 2 ^ n ) )  e. 
RR* )
29 icossre 11605 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  /  (
2 ^ n ) )  e.  RR  /\  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )  ->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) )  C_  RR )
3023, 28, 29syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  C_  RR )
3112, 30eqsstrd 3538 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  d  C_  RR )
3231ex 434 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) )  ->  d  C_  RR ) )
3332rexlimivv 2960 . . . . . . . . . . . . 13  |-  ( E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  ->  d  C_  RR )
3411, 33sylbi 195 . . . . . . . . . . . 12  |-  ( d  e.  ran  I  -> 
d  C_  RR )
358, 34mprgbir 2828 . . . . . . . . . . 11  |-  ran  I  C_ 
~P RR
3635sseli 3500 . . . . . . . . . 10  |-  ( u  e.  ran  I  ->  u  e.  ~P RR )
3736elpwid 4020 . . . . . . . . 9  |-  ( u  e.  ran  I  ->  u  C_  RR )
3835sseli 3500 . . . . . . . . . 10  |-  ( v  e.  ran  I  -> 
v  e.  ~P RR )
3938elpwid 4020 . . . . . . . . 9  |-  ( v  e.  ran  I  -> 
v  C_  RR )
40 xpss12 5108 . . . . . . . . 9  |-  ( ( u  C_  RR  /\  v  C_  RR )  ->  (
u  X.  v ) 
C_  ( RR  X.  RR ) )
4137, 39, 40syl2an 477 . . . . . . . 8  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
u  X.  v ) 
C_  ( RR  X.  RR ) )
4241adantr 465 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
( u  X.  v
)  C_  ( RR  X.  RR ) )
437, 42eqsstrd 3538 . . . . . 6  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  C_  ( RR  X.  RR ) )
4443ex 434 . . . . 5  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
d  =  ( u  X.  v )  -> 
d  C_  ( RR  X.  RR ) ) )
4544rexlimivv 2960 . . . 4  |-  ( E. u  e.  ran  I E. v  e.  ran  I  d  =  (
u  X.  v )  ->  d  C_  ( RR  X.  RR ) )
466, 45sylbi 195 . . 3  |-  ( d  e.  ran  R  -> 
d  C_  ( RR  X.  RR ) )
471, 46mprgbir 2828 . 2  |-  U. ran  R 
C_  ( RR  X.  RR )
48 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
49 retop 21031 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
5048, 49eqeltri 2551 . . . . 5  |-  J  e. 
Top
5150, 50txtopi 19854 . . . 4  |-  ( J 
tX  J )  e. 
Top
52 uniretop 21032 . . . . . . 7  |-  RR  =  U. ( topGen `  ran  (,) )
5348unieqi 4254 . . . . . . 7  |-  U. J  =  U. ( topGen `  ran  (,) )
5452, 53eqtr4i 2499 . . . . . 6  |-  RR  =  U. J
5550, 50, 54, 54txunii 19857 . . . . 5  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
5655topopn 19210 . . . 4  |-  ( ( J  tX  J )  e.  Top  ->  ( RR  X.  RR )  e.  ( J  tX  J
) )
5748, 9, 2dya2iocuni 27922 . . . 4  |-  ( ( RR  X.  RR )  e.  ( J  tX  J )  ->  E. c  e.  ~P  ran  R U. c  =  ( RR  X.  RR ) )
5851, 56, 57mp2b 10 . . 3  |-  E. c  e.  ~P  ran  R U. c  =  ( RR  X.  RR )
59 simpr 461 . . . . 5  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  U. c  =  ( RR  X.  RR ) )
60 elpwi 4019 . . . . . . 7  |-  ( c  e.  ~P ran  R  ->  c  C_  ran  R )
6160adantr 465 . . . . . 6  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  c  C_ 
ran  R )
6261unissd 4269 . . . . 5  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  U. c  C_ 
U. ran  R )
6359, 62eqsstr3d 3539 . . . 4  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  ( RR  X.  RR )  C_  U.
ran  R )
6463rexlimiva 2951 . . 3  |-  ( E. c  e.  ~P  ran  R U. c  =  ( RR  X.  RR )  ->  ( RR  X.  RR )  C_  U. ran  R )
6558, 64ax-mp 5 . 2  |-  ( RR 
X.  RR )  C_  U.
ran  R
6647, 65eqssi 3520 1  |-  U. ran  R  =  ( RR  X.  RR )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    C_ wss 3476   ~Pcpw 4010   U.cuni 4245    X. cxp 4997   ran crn 5000   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495   RR*cxr 9627    / cdiv 10206   2c2 10585   ZZcz 10864   (,)cioo 11529   [,)cico 11531   ^cexp 12134   topGenctg 14693   Topctop 19189    tX ctx 19824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-fac 12322  df-bc 12349  df-hash 12374  df-shft 12863  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-limsup 13257  df-clim 13274  df-rlim 13275  df-sum 13472  df-ef 13665  df-sin 13667  df-cos 13668  df-pi 13670  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-refld 18436  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-lp 19431  df-perf 19432  df-cn 19522  df-cnp 19523  df-haus 19610  df-cmp 19681  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-fcls 20205  df-xms 20586  df-ms 20587  df-tms 20588  df-cncf 21145  df-cfil 21457  df-cmet 21459  df-cms 21537  df-limc 22033  df-dv 22034  df-log 22700  df-cxp 22701  df-logb 27675
This theorem is referenced by:  sxbrsigalem1  27924  sxbrsigalem2  27925  sxbrsigalem5  27927
  Copyright terms: Public domain W3C validator