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

Theorem dya2iocucvr 29179
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 4221 . . 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 3034 . . . . . 6  |-  u  e. 
_V
4 vex 3034 . . . . . 6  |-  v  e. 
_V
53, 4xpex 6614 . . . . 5  |-  ( u  X.  v )  e. 
_V
62, 5elrnmpt2 6428 . . . 4  |-  ( d  e.  ran  R  <->  E. u  e.  ran  I E. v  e.  ran  I  d  =  ( u  X.  v
) )
7 simpr 468 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  =  ( u  X.  v ) )
8 pwssb 4361 . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . 14  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
119, 10elrnmpt2 6428 . . . . . . . . . . . . 13  |-  ( d  e.  ran  I  <->  E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )
12 simpr 468 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  x  e.  ZZ )
1413zred 11063 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  x  e.  RR )
15 2re 10701 . . . . . . . . . . . . . . . . . . . 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 10724 . . . . . . . . . . . . . . . . . . . 20  |-  2  =/=  0
1817a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  2  =/=  0 )
19 simplr 770 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  n  e.  ZZ )
2016, 18, 19reexpclzd 12479 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( 2 ^ n )  e.  RR )
21 2cnd 10704 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  2  e.  CC )
2221, 18, 19expne0d 12460 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( 2 ^ n )  =/=  0 )
2314, 20, 22redivcld 10457 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( x  /  ( 2 ^ n ) )  e.  RR )
24 1red 9676 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  1  e.  RR )
2514, 24readdcld 9688 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( x  +  1 )  e.  RR )
2625, 20, 22redivcld 10457 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  +  1 )  /  ( 2 ^ n ) )  e.  RR )
2726rexrd 9708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  ( (
x  +  1 )  /  ( 2 ^ n ) )  e. 
RR* )
28 icossre 11740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  /  (
2 ^ n ) )  e.  RR  /\  ( ( x  + 
1 )  /  (
2 ^ n ) )  e.  RR* )  ->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) )  C_  RR )
2923, 27, 28syl2anc 673 . . . . . . . . . . . . . . . 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 3452 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  ZZ  /\  n  e.  ZZ )  /\  d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) ) )  ->  d  C_  RR )
3130ex 441 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ZZ  /\  n  e.  ZZ )  ->  ( d  =  ( ( x  /  (
2 ^ n ) ) [,) ( ( x  +  1 )  /  ( 2 ^ n ) ) )  ->  d  C_  RR ) )
3231rexlimivv 2876 . . . . . . . . . . . . 13  |-  ( E. x  e.  ZZ  E. n  e.  ZZ  d  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  ->  d  C_  RR )
3311, 32sylbi 200 . . . . . . . . . . . 12  |-  ( d  e.  ran  I  -> 
d  C_  RR )
348, 33mprgbir 2771 . . . . . . . . . . 11  |-  ran  I  C_ 
~P RR
3534sseli 3414 . . . . . . . . . 10  |-  ( u  e.  ran  I  ->  u  e.  ~P RR )
3635elpwid 3952 . . . . . . . . 9  |-  ( u  e.  ran  I  ->  u  C_  RR )
3734sseli 3414 . . . . . . . . . 10  |-  ( v  e.  ran  I  -> 
v  e.  ~P RR )
3837elpwid 3952 . . . . . . . . 9  |-  ( v  e.  ran  I  -> 
v  C_  RR )
39 xpss12 4945 . . . . . . . . 9  |-  ( ( u  C_  RR  /\  v  C_  RR )  ->  (
u  X.  v ) 
C_  ( RR  X.  RR ) )
4036, 38, 39syl2an 485 . . . . . . . 8  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
u  X.  v ) 
C_  ( RR  X.  RR ) )
4140adantr 472 . . . . . . 7  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
( u  X.  v
)  C_  ( RR  X.  RR ) )
427, 41eqsstrd 3452 . . . . . 6  |-  ( ( ( u  e.  ran  I  /\  v  e.  ran  I )  /\  d  =  ( u  X.  v ) )  -> 
d  C_  ( RR  X.  RR ) )
4342ex 441 . . . . 5  |-  ( ( u  e.  ran  I  /\  v  e.  ran  I )  ->  (
d  =  ( u  X.  v )  -> 
d  C_  ( RR  X.  RR ) ) )
4443rexlimivv 2876 . . . 4  |-  ( E. u  e.  ran  I E. v  e.  ran  I  d  =  (
u  X.  v )  ->  d  C_  ( RR  X.  RR ) )
456, 44sylbi 200 . . 3  |-  ( d  e.  ran  R  -> 
d  C_  ( RR  X.  RR ) )
461, 45mprgbir 2771 . 2  |-  U. ran  R 
C_  ( RR  X.  RR )
47 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
48 retop 21860 . . . . . 6  |-  ( topGen ` 
ran  (,) )  e.  Top
4947, 48eqeltri 2545 . . . . 5  |-  J  e. 
Top
5049, 49txtopi 20682 . . . 4  |-  ( J 
tX  J )  e. 
Top
51 uniretop 21861 . . . . . . 7  |-  RR  =  U. ( topGen `  ran  (,) )
5247unieqi 4199 . . . . . . 7  |-  U. J  =  U. ( topGen `  ran  (,) )
5351, 52eqtr4i 2496 . . . . . 6  |-  RR  =  U. J
5449, 49, 53, 53txunii 20685 . . . . 5  |-  ( RR 
X.  RR )  = 
U. ( J  tX  J )
5554topopn 20013 . . . 4  |-  ( ( J  tX  J )  e.  Top  ->  ( RR  X.  RR )  e.  ( J  tX  J
) )
5647, 9, 2dya2iocuni 29178 . . . 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 468 . . . . 5  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  U. c  =  ( RR  X.  RR ) )
59 elpwi 3951 . . . . . . 7  |-  ( c  e.  ~P ran  R  ->  c  C_  ran  R )
6059adantr 472 . . . . . 6  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  c  C_ 
ran  R )
6160unissd 4214 . . . . 5  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  U. c  C_ 
U. ran  R )
6258, 61eqsstr3d 3453 . . . 4  |-  ( ( c  e.  ~P ran  R  /\  U. c  =  ( RR  X.  RR ) )  ->  ( RR  X.  RR )  C_  U.
ran  R )
6362rexlimiva 2868 . . 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 3434 1  |-  U. ran  R  =  ( RR  X.  RR )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757    C_ wss 3390   ~Pcpw 3942   U.cuni 4190    X. cxp 4837   ran crn 4840   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560   RR*cxr 9692    / cdiv 10291   2c2 10681   ZZcz 10961   (,)cioo 11660   [,)cico 11662   ^cexp 12310   topGenctg 15414   Topctop 19994    tX ctx 20652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-refld 19250  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-fcls 21034  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-cfil 22303  df-cmet 22305  df-cms 22381  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-logb 23781
This theorem is referenced by:  sxbrsigalem1  29180  sxbrsigalem2  29181  sxbrsigalem5  29183
  Copyright terms: Public domain W3C validator