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Theorem dyaddisj 21056
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.)
Hypothesis
Ref Expression
dyadmbl.1  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
Assertion
Ref Expression
dyaddisj  |-  ( ( A  e.  ran  F  /\  B  e.  ran  F )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  \/  ( [,] `  B
)  C_  ( [,] `  A )  \/  (
( (,) `  A
)  i^i  ( (,) `  B ) )  =  (/) ) )
Distinct variable groups:    x, y, B    x, A, y    x, F, y

Proof of Theorem dyaddisj
Dummy variables  c 
d  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dyadmbl.1 . . . . 5  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
21dyadf 21051 . . . 4  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
3 ffn 5554 . . . 4  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  ( ZZ  X.  NN0 ) )
4 ovelrn 6234 . . . . 5  |-  ( F  Fn  ( ZZ  X.  NN0 )  ->  ( A  e.  ran  F  <->  E. a  e.  ZZ  E. c  e. 
NN0  A  =  (
a F c ) ) )
5 ovelrn 6234 . . . . 5  |-  ( F  Fn  ( ZZ  X.  NN0 )  ->  ( B  e.  ran  F  <->  E. b  e.  ZZ  E. d  e. 
NN0  B  =  (
b F d ) ) )
64, 5anbi12d 710 . . . 4  |-  ( F  Fn  ( ZZ  X.  NN0 )  ->  ( ( A  e.  ran  F  /\  B  e.  ran  F )  <->  ( E. a  e.  ZZ  E. c  e. 
NN0  A  =  (
a F c )  /\  E. b  e.  ZZ  E. d  e. 
NN0  B  =  (
b F d ) ) ) )
72, 3, 6mp2b 10 . . 3  |-  ( ( A  e.  ran  F  /\  B  e.  ran  F )  <->  ( E. a  e.  ZZ  E. c  e. 
NN0  A  =  (
a F c )  /\  E. b  e.  ZZ  E. d  e. 
NN0  B  =  (
b F d ) ) )
8 reeanv 2883 . . 3  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  ( E. c  e.  NN0  A  =  ( a F c )  /\  E. d  e.  NN0  B  =  ( b F d ) )  <->  ( E. a  e.  ZZ  E. c  e.  NN0  A  =  ( a F c )  /\  E. b  e.  ZZ  E. d  e. 
NN0  B  =  (
b F d ) ) )
97, 8bitr4i 252 . 2  |-  ( ( A  e.  ran  F  /\  B  e.  ran  F )  <->  E. a  e.  ZZ  E. b  e.  ZZ  ( E. c  e.  NN0  A  =  ( a F c )  /\  E. d  e.  NN0  B  =  ( b F d ) ) )
10 reeanv 2883 . . . 4  |-  ( E. c  e.  NN0  E. d  e.  NN0  ( A  =  ( a F c )  /\  B  =  ( b F d ) )  <->  ( E. c  e.  NN0  A  =  ( a F c )  /\  E. d  e.  NN0  B  =  ( b F d ) ) )
11 nn0re 10580 . . . . . . . 8  |-  ( c  e.  NN0  ->  c  e.  RR )
1211ad2antrl 727 . . . . . . 7  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  ->  c  e.  RR )
13 nn0re 10580 . . . . . . . 8  |-  ( d  e.  NN0  ->  d  e.  RR )
1413ad2antll 728 . . . . . . 7  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  ->  d  e.  RR )
151dyaddisjlem 21055 . . . . . . 7  |-  ( ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  (
c  e.  NN0  /\  d  e.  NN0 ) )  /\  c  <_  d
)  ->  ( ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) )  \/  (
( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) ) )
16 ancom 450 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  <->  ( b  e.  ZZ  /\  a  e.  ZZ )
)
17 ancom 450 . . . . . . . . . 10  |-  ( ( c  e.  NN0  /\  d  e.  NN0 )  <->  ( d  e.  NN0  /\  c  e. 
NN0 ) )
1816, 17anbi12i 697 . . . . . . . . 9  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  <->  ( (
b  e.  ZZ  /\  a  e.  ZZ )  /\  ( d  e.  NN0  /\  c  e.  NN0 )
) )
191dyaddisjlem 21055 . . . . . . . . 9  |-  ( ( ( ( b  e.  ZZ  /\  a  e.  ZZ )  /\  (
d  e.  NN0  /\  c  e.  NN0 ) )  /\  d  <_  c
)  ->  ( ( [,] `  ( b F d ) )  C_  ( [,] `  ( a F c ) )  \/  ( [,] `  (
a F c ) )  C_  ( [,] `  ( b F d ) )  \/  (
( (,) `  (
b F d ) )  i^i  ( (,) `  ( a F c ) ) )  =  (/) ) )
2018, 19sylanb 472 . . . . . . . 8  |-  ( ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  (
c  e.  NN0  /\  d  e.  NN0 ) )  /\  d  <_  c
)  ->  ( ( [,] `  ( b F d ) )  C_  ( [,] `  ( a F c ) )  \/  ( [,] `  (
a F c ) )  C_  ( [,] `  ( b F d ) )  \/  (
( (,) `  (
b F d ) )  i^i  ( (,) `  ( a F c ) ) )  =  (/) ) )
21 orcom 387 . . . . . . . . . 10  |-  ( ( ( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) )  \/  ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) ) )  <->  ( ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) ) ) )
22 incom 3538 . . . . . . . . . . 11  |-  ( ( (,) `  ( b F d ) )  i^i  ( (,) `  (
a F c ) ) )  =  ( ( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )
2322eqeq1i 2445 . . . . . . . . . 10  |-  ( ( ( (,) `  (
b F d ) )  i^i  ( (,) `  ( a F c ) ) )  =  (/) 
<->  ( ( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) )
2421, 23orbi12i 521 . . . . . . . . 9  |-  ( ( ( ( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) )  \/  ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) ) )  \/  ( ( (,) `  ( b F d ) )  i^i  ( (,) `  (
a F c ) ) )  =  (/) ) 
<->  ( ( ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) ) )  \/  ( ( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) ) )
25 df-3or 966 . . . . . . . . 9  |-  ( ( ( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) )  \/  ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( ( (,) `  ( b F d ) )  i^i  ( (,) `  ( a F c ) ) )  =  (/) )  <->  ( (
( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) )  \/  ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) ) )  \/  ( ( (,) `  ( b F d ) )  i^i  ( (,) `  (
a F c ) ) )  =  (/) ) )
26 df-3or 966 . . . . . . . . 9  |-  ( ( ( [,] `  (
a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( [,] `  ( b F d ) )  C_  ( [,] `  ( a F c ) )  \/  ( ( (,) `  ( a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) )  <->  ( (
( [,] `  (
a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( [,] `  ( b F d ) )  C_  ( [,] `  ( a F c ) ) )  \/  ( ( (,) `  ( a F c ) )  i^i  ( (,) `  (
b F d ) ) )  =  (/) ) )
2724, 25, 263bitr4i 277 . . . . . . . 8  |-  ( ( ( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) )  \/  ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( ( (,) `  ( b F d ) )  i^i  ( (,) `  ( a F c ) ) )  =  (/) )  <->  ( ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) )  \/  (
( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) ) )
2820, 27sylib 196 . . . . . . 7  |-  ( ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  (
c  e.  NN0  /\  d  e.  NN0 ) )  /\  d  <_  c
)  ->  ( ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) )  \/  (
( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) ) )
2912, 14, 15, 28lecasei 9472 . . . . . 6  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  ->  (
( [,] `  (
a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( [,] `  ( b F d ) )  C_  ( [,] `  ( a F c ) )  \/  ( ( (,) `  ( a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) ) )
30 simpl 457 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  A  =  ( a F c ) )
3130fveq2d 5690 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( [,] `  A
)  =  ( [,] `  ( a F c ) ) )
32 simpr 461 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  B  =  ( b F d ) )
3332fveq2d 5690 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( [,] `  B
)  =  ( [,] `  ( b F d ) ) )
3431, 33sseq12d 3380 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  <->  ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) ) ) )
3533, 31sseq12d 3380 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  B )  C_  ( [,] `  A )  <->  ( [,] `  ( b F d ) )  C_  ( [,] `  ( a F c ) ) ) )
3630fveq2d 5690 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( (,) `  A
)  =  ( (,) `  ( a F c ) ) )
3732fveq2d 5690 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( (,) `  B
)  =  ( (,) `  ( b F d ) ) )
3836, 37ineq12d 3548 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( (,) `  A )  i^i  ( (,) `  B ) )  =  ( ( (,) `  ( a F c ) )  i^i  ( (,) `  ( b F d ) ) ) )
3938eqeq1d 2446 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( ( (,) `  A )  i^i  ( (,) `  B
) )  =  (/)  <->  (
( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) ) )
4034, 35, 393orbi123d 1288 . . . . . 6  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( ( [,] `  A ) 
C_  ( [,] `  B
)  \/  ( [,] `  B )  C_  ( [,] `  A )  \/  ( ( (,) `  A
)  i^i  ( (,) `  B ) )  =  (/) )  <->  ( ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) )  \/  ( [,] `  (
b F d ) )  C_  ( [,] `  ( a F c ) )  \/  (
( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) ) ) )
4129, 40syl5ibrcom 222 . . . . 5  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  ->  (
( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  \/  ( [,] `  B
)  C_  ( [,] `  A )  \/  (
( (,) `  A
)  i^i  ( (,) `  B ) )  =  (/) ) ) )
4241rexlimdvva 2843 . . . 4  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( E. c  e. 
NN0  E. d  e.  NN0  ( A  =  (
a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  \/  ( [,] `  B
)  C_  ( [,] `  A )  \/  (
( (,) `  A
)  i^i  ( (,) `  B ) )  =  (/) ) ) )
4310, 42syl5bir 218 . . 3  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( E. c  e.  NN0  A  =  ( a F c )  /\  E. d  e. 
NN0  B  =  (
b F d ) )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  \/  ( [,] `  B
)  C_  ( [,] `  A )  \/  (
( (,) `  A
)  i^i  ( (,) `  B ) )  =  (/) ) ) )
4443rexlimivv 2841 . 2  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  ( E. c  e.  NN0  A  =  ( a F c )  /\  E. d  e.  NN0  B  =  ( b F d ) )  ->  (
( [,] `  A
)  C_  ( [,] `  B )  \/  ( [,] `  B )  C_  ( [,] `  A )  \/  ( ( (,) `  A )  i^i  ( (,) `  B ) )  =  (/) ) )
459, 44sylbi 195 1  |-  ( ( A  e.  ran  F  /\  B  e.  ran  F )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  \/  ( [,] `  B
)  C_  ( [,] `  A )  \/  (
( (,) `  A
)  i^i  ( (,) `  B ) )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 964    = wceq 1369    e. wcel 1756   E.wrex 2711    i^i cin 3322    C_ wss 3323   (/)c0 3632   <.cop 3878   class class class wbr 4287    X. cxp 4833   ran crn 4836    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   RRcr 9273   1c1 9275    + caddc 9277    <_ cle 9411    / cdiv 9985   2c2 10363   NN0cn0 10571   ZZcz 10638   (,)cioo 11292   [,]cicc 11295   ^cexp 11857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-ioo 11296  df-icc 11299  df-seq 11799  df-exp 11858
This theorem is referenced by:  dyadmbl  21060  mblfinlem2  28400
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