MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dyadss Unicode version

Theorem dyadss 18949
Description: Two closed dyadic rational intervals are either in a subset relationship or are almost disjoint (the interiors are disjoint). (Contributed by Mario Carneiro, 26-Mar-2015.) (Proof shortened by Mario Carneiro, 26-Apr-2016.)
Hypothesis
Ref Expression
dyadmbl.1  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
Assertion
Ref Expression
dyadss  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e. 
NN0  /\  D  e.  NN0 ) )  ->  (
( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) )  ->  D  <_  C
) )
Distinct variable groups:    x, y, B    x, C, y    x, A, y    x, D, y   
x, F, y

Proof of Theorem dyadss
StepHypRef Expression
1 simpr 447 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( [,] `  ( A F C ) )  C_  ( [,] `  ( B F D ) ) )
2 simpllr 735 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  B  e.  ZZ )
3 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  D  e.  NN0 )
4 dyadmbl.1 . . . . . . . . . . 11  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
54dyadval 18947 . . . . . . . . . 10  |-  ( ( B  e.  ZZ  /\  D  e.  NN0 )  -> 
( B F D )  =  <. ( B  /  ( 2 ^ D ) ) ,  ( ( B  + 
1 )  /  (
2 ^ D ) ) >. )
62, 3, 5syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( B F D )  =  <. ( B  /  ( 2 ^ D ) ) ,  ( ( B  +  1 )  / 
( 2 ^ D
) ) >. )
76fveq2d 5529 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( [,] `  ( B F D ) )  =  ( [,] `  <. ( B  /  ( 2 ^ D ) ) ,  ( ( B  + 
1 )  /  (
2 ^ D ) ) >. ) )
8 df-ov 5861 . . . . . . . 8  |-  ( ( B  /  ( 2 ^ D ) ) [,] ( ( B  +  1 )  / 
( 2 ^ D
) ) )  =  ( [,] `  <. ( B  /  ( 2 ^ D ) ) ,  ( ( B  +  1 )  / 
( 2 ^ D
) ) >. )
97, 8syl6eqr 2333 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( [,] `  ( B F D ) )  =  ( ( B  /  (
2 ^ D ) ) [,] ( ( B  +  1 )  /  ( 2 ^ D ) ) ) )
102zred 10117 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  B  e.  RR )
11 2nn 9877 . . . . . . . . . 10  |-  2  e.  NN
12 nnexpcl 11116 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  D  e.  NN0 )  -> 
( 2 ^ D
)  e.  NN )
1311, 3, 12sylancr 644 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( 2 ^ D )  e.  NN )
1410, 13nndivred 9794 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( B  /  ( 2 ^ D ) )  e.  RR )
15 peano2re 8985 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  +  1 )  e.  RR )
1610, 15syl 15 . . . . . . . . 9  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( B  +  1 )  e.  RR )
1716, 13nndivred 9794 . . . . . . . 8  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( ( B  +  1 )  /  ( 2 ^ D ) )  e.  RR )
18 iccssre 10731 . . . . . . . 8  |-  ( ( ( B  /  (
2 ^ D ) )  e.  RR  /\  ( ( B  + 
1 )  /  (
2 ^ D ) )  e.  RR )  ->  ( ( B  /  ( 2 ^ D ) ) [,] ( ( B  + 
1 )  /  (
2 ^ D ) ) )  C_  RR )
1914, 17, 18syl2anc 642 . . . . . . 7  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( ( B  /  ( 2 ^ D ) ) [,] ( ( B  + 
1 )  /  (
2 ^ D ) ) )  C_  RR )
209, 19eqsstrd 3212 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( [,] `  ( B F D ) )  C_  RR )
21 ovolss 18844 . . . . . 6  |-  ( ( ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) )  /\  ( [,] `  ( B F D ) ) 
C_  RR )  -> 
( vol * `  ( [,] `  ( A F C ) ) )  <_  ( vol * `
 ( [,] `  ( B F D ) ) ) )
221, 20, 21syl2anc 642 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( vol * `
 ( [,] `  ( A F C ) ) )  <_  ( vol * `
 ( [,] `  ( B F D ) ) ) )
23 simplll 734 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  A  e.  ZZ )
24 simplrl 736 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  C  e.  NN0 )
254dyadovol 18948 . . . . . 6  |-  ( ( A  e.  ZZ  /\  C  e.  NN0 )  -> 
( vol * `  ( [,] `  ( A F C ) ) )  =  ( 1  /  ( 2 ^ C ) ) )
2623, 24, 25syl2anc 642 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( vol * `
 ( [,] `  ( A F C ) ) )  =  ( 1  /  ( 2 ^ C ) ) )
274dyadovol 18948 . . . . . 6  |-  ( ( B  e.  ZZ  /\  D  e.  NN0 )  -> 
( vol * `  ( [,] `  ( B F D ) ) )  =  ( 1  /  ( 2 ^ D ) ) )
282, 3, 27syl2anc 642 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( vol * `
 ( [,] `  ( B F D ) ) )  =  ( 1  /  ( 2 ^ D ) ) )
2922, 26, 283brtr3d 4052 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( 1  /  ( 2 ^ C ) )  <_ 
( 1  /  (
2 ^ D ) ) )
30 nnexpcl 11116 . . . . . 6  |-  ( ( 2  e.  NN  /\  C  e.  NN0 )  -> 
( 2 ^ C
)  e.  NN )
3111, 24, 30sylancr 644 . . . . 5  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( 2 ^ C )  e.  NN )
32 nnre 9753 . . . . . . 7  |-  ( ( 2 ^ D )  e.  NN  ->  (
2 ^ D )  e.  RR )
33 nngt0 9775 . . . . . . 7  |-  ( ( 2 ^ D )  e.  NN  ->  0  <  ( 2 ^ D
) )
3432, 33jca 518 . . . . . 6  |-  ( ( 2 ^ D )  e.  NN  ->  (
( 2 ^ D
)  e.  RR  /\  0  <  ( 2 ^ D ) ) )
35 nnre 9753 . . . . . . 7  |-  ( ( 2 ^ C )  e.  NN  ->  (
2 ^ C )  e.  RR )
36 nngt0 9775 . . . . . . 7  |-  ( ( 2 ^ C )  e.  NN  ->  0  <  ( 2 ^ C
) )
3735, 36jca 518 . . . . . 6  |-  ( ( 2 ^ C )  e.  NN  ->  (
( 2 ^ C
)  e.  RR  /\  0  <  ( 2 ^ C ) ) )
38 lerec 9638 . . . . . 6  |-  ( ( ( ( 2 ^ D )  e.  RR  /\  0  <  ( 2 ^ D ) )  /\  ( ( 2 ^ C )  e.  RR  /\  0  < 
( 2 ^ C
) ) )  -> 
( ( 2 ^ D )  <_  (
2 ^ C )  <-> 
( 1  /  (
2 ^ C ) )  <_  ( 1  /  ( 2 ^ D ) ) ) )
3934, 37, 38syl2an 463 . . . . 5  |-  ( ( ( 2 ^ D
)  e.  NN  /\  ( 2 ^ C
)  e.  NN )  ->  ( ( 2 ^ D )  <_ 
( 2 ^ C
)  <->  ( 1  / 
( 2 ^ C
) )  <_  (
1  /  ( 2 ^ D ) ) ) )
4013, 31, 39syl2anc 642 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( (
2 ^ D )  <_  ( 2 ^ C )  <->  ( 1  /  ( 2 ^ C ) )  <_ 
( 1  /  (
2 ^ D ) ) ) )
4129, 40mpbird 223 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( 2 ^ D )  <_ 
( 2 ^ C
) )
42 2re 9815 . . . . 5  |-  2  e.  RR
4342a1i 10 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  2  e.  RR )
443nn0zd 10115 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  D  e.  ZZ )
4524nn0zd 10115 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  C  e.  ZZ )
46 1lt2 9886 . . . . 5  |-  1  <  2
4746a1i 10 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  1  <  2 )
4843, 44, 45, 47leexp2d 11275 . . 3  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  ( D  <_  C  <->  ( 2 ^ D )  <_  (
2 ^ C ) ) )
4941, 48mpbird 223 . 2  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  D  <_  C )
5049ex 423 1  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e. 
NN0  /\  D  e.  NN0 ) )  ->  (
( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) )  ->  D  <_  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   [,]cicc 10659   ^cexp 11104   vol *covol 18822
This theorem is referenced by:  dyadmaxlem  18952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cmp 17114  df-ovol 18824
  Copyright terms: Public domain W3C validator