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Theorem dyadss 19486
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 448 . . . . . 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 736 . . . . . . . . . 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 738 . . . . . . . . . 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 19484 . . . . . . . . . 10  |-  ( ( B  e.  ZZ  /\  D  e.  NN0 )  -> 
( B F D )  =  <. ( B  /  ( 2 ^ D ) ) ,  ( ( B  + 
1 )  /  (
2 ^ D ) ) >. )
62, 3, 5syl2anc 643 . . . . . . . . 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 5732 . . . . . . . 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 6084 . . . . . . . 8  |-  ( ( B  /  ( 2 ^ D ) ) [,] ( ( B  +  1 )  / 
( 2 ^ D
) ) )  =  ( [,] `  <. ( B  /  ( 2 ^ D ) ) ,  ( ( B  +  1 )  / 
( 2 ^ D
) ) >. )
97, 8syl6eqr 2486 . . . . . . 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 10375 . . . . . . . . 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 10133 . . . . . . . . . 10  |-  2  e.  NN
12 nnexpcl 11394 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  D  e.  NN0 )  -> 
( 2 ^ D
)  e.  NN )
1311, 3, 12sylancr 645 . . . . . . . . 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 10048 . . . . . . . 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 9239 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  +  1 )  e.  RR )
1610, 15syl 16 . . . . . . . . 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 10048 . . . . . . . 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 10992 . . . . . . . 8  |-  ( ( ( B  /  (
2 ^ D ) )  e.  RR  /\  ( ( B  + 
1 )  /  (
2 ^ D ) )  e.  RR )  ->  ( ( B  /  ( 2 ^ D ) ) [,] ( ( B  + 
1 )  /  (
2 ^ D ) ) )  C_  RR )
1914, 17, 18syl2anc 643 . . . . . . 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 3382 . . . . . 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 19381 . . . . . 6  |-  ( ( ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) )  /\  ( [,] `  ( B F D ) ) 
C_  RR )  -> 
( vol * `  ( [,] `  ( A F C ) ) )  <_  ( vol * `
 ( [,] `  ( B F D ) ) ) )
221, 20, 21syl2anc 643 . . . . 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 735 . . . . . 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 737 . . . . . 6  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  C  e.  NN0 )
254dyadovol 19485 . . . . . 6  |-  ( ( A  e.  ZZ  /\  C  e.  NN0 )  -> 
( vol * `  ( [,] `  ( A F C ) ) )  =  ( 1  /  ( 2 ^ C ) ) )
2623, 24, 25syl2anc 643 . . . . 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 19485 . . . . . 6  |-  ( ( B  e.  ZZ  /\  D  e.  NN0 )  -> 
( vol * `  ( [,] `  ( B F D ) ) )  =  ( 1  /  ( 2 ^ D ) ) )
282, 3, 27syl2anc 643 . . . . 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 4241 . . . 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 11394 . . . . . 6  |-  ( ( 2  e.  NN  /\  C  e.  NN0 )  -> 
( 2 ^ C
)  e.  NN )
3111, 24, 30sylancr 645 . . . . 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 10007 . . . . . . 7  |-  ( ( 2 ^ D )  e.  NN  ->  (
2 ^ D )  e.  RR )
33 nngt0 10029 . . . . . . 7  |-  ( ( 2 ^ D )  e.  NN  ->  0  <  ( 2 ^ D
) )
3432, 33jca 519 . . . . . 6  |-  ( ( 2 ^ D )  e.  NN  ->  (
( 2 ^ D
)  e.  RR  /\  0  <  ( 2 ^ D ) ) )
35 nnre 10007 . . . . . . 7  |-  ( ( 2 ^ C )  e.  NN  ->  (
2 ^ C )  e.  RR )
36 nngt0 10029 . . . . . . 7  |-  ( ( 2 ^ C )  e.  NN  ->  0  <  ( 2 ^ C
) )
3735, 36jca 519 . . . . . 6  |-  ( ( 2 ^ C )  e.  NN  ->  (
( 2 ^ C
)  e.  RR  /\  0  <  ( 2 ^ C ) ) )
38 lerec 9892 . . . . . 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 464 . . . . 5  |-  ( ( ( 2 ^ D
)  e.  NN  /\  ( 2 ^ C
)  e.  NN )  ->  ( ( 2 ^ D )  <_ 
( 2 ^ C
)  <->  ( 1  / 
( 2 ^ C
) )  <_  (
1  /  ( 2 ^ D ) ) ) )
4013, 31, 39syl2anc 643 . . . 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 224 . . 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 10069 . . . . 5  |-  2  e.  RR
4342a1i 11 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  2  e.  RR )
443nn0zd 10373 . . . 4  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  D  e.  ZZ )
4524nn0zd 10373 . . . 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 10142 . . . . 5  |-  1  <  2
4746a1i 11 . . . 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 11553 . . 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 224 . 2  |-  ( ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( C  e.  NN0  /\  D  e.  NN0 ) )  /\  ( [,] `  ( A F C ) ) 
C_  ( [,] `  ( B F D ) ) )  ->  D  <_  C )
5049ex 424 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320   <.cop 3817   class class class wbr 4212   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    <_ cle 9121    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   [,]cicc 10919   ^cexp 11382   vol *covol 19359
This theorem is referenced by:  dyadmaxlem  19489
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-rest 13650  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-top 16963  df-bases 16965  df-topon 16966  df-cmp 17450  df-ovol 19361
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