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Theorem dyaddisj 19441
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 19436 . . . 4  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
3 ffn 5550 . . . 4  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  ( ZZ  X.  NN0 ) )
4 ovelrn 6181 . . . . 5  |-  ( F  Fn  ( ZZ  X.  NN0 )  ->  ( A  e.  ran  F  <->  E. a  e.  ZZ  E. c  e. 
NN0  A  =  (
a F c ) ) )
5 ovelrn 6181 . . . . 5  |-  ( F  Fn  ( ZZ  X.  NN0 )  ->  ( B  e.  ran  F  <->  E. b  e.  ZZ  E. d  e. 
NN0  B  =  (
b F d ) ) )
64, 5anbi12d 692 . . . 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 2835 . . 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 244 . 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 2835 . . . 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 10186 . . . . . . . 8  |-  ( c  e.  NN0  ->  c  e.  RR )
1211ad2antrl 709 . . . . . . 7  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  ->  c  e.  RR )
13 nn0re 10186 . . . . . . . 8  |-  ( d  e.  NN0  ->  d  e.  RR )
1413ad2antll 710 . . . . . . 7  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  ->  d  e.  RR )
151dyaddisjlem 19440 . . . . . . 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 438 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  <->  ( b  e.  ZZ  /\  a  e.  ZZ )
)
17 ancom 438 . . . . . . . . . 10  |-  ( ( c  e.  NN0  /\  d  e.  NN0 )  <->  ( d  e.  NN0  /\  c  e. 
NN0 ) )
1816, 17anbi12i 679 . . . . . . . . 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 19440 . . . . . . . . 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 459 . . . . . . . 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 377 . . . . . . . . . 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 3493 . . . . . . . . . . 11  |-  ( ( (,) `  ( b F d ) )  i^i  ( (,) `  (
a F c ) ) )  =  ( ( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )
2322eqeq1i 2411 . . . . . . . . . 10  |-  ( ( ( (,) `  (
b F d ) )  i^i  ( (,) `  ( a F c ) ) )  =  (/) 
<->  ( ( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) )
2421, 23orbi12i 508 . . . . . . . . 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 937 . . . . . . . . 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 937 . . . . . . . . 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 269 . . . . . . . 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 189 . . . . . . 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 9135 . . . . . 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 444 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  A  =  ( a F c ) )
3130fveq2d 5691 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( [,] `  A
)  =  ( [,] `  ( a F c ) ) )
32 simpr 448 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  B  =  ( b F d ) )
3332fveq2d 5691 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( [,] `  B
)  =  ( [,] `  ( b F d ) ) )
3431, 33sseq12d 3337 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  <->  ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) ) ) )
3533, 31sseq12d 3337 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  B )  C_  ( [,] `  A )  <->  ( [,] `  ( b F d ) )  C_  ( [,] `  ( a F c ) ) ) )
3630fveq2d 5691 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( (,) `  A
)  =  ( (,) `  ( a F c ) ) )
3732fveq2d 5691 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( (,) `  B
)  =  ( (,) `  ( b F d ) ) )
3836, 37ineq12d 3503 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( (,) `  A )  i^i  ( (,) `  B ) )  =  ( ( (,) `  ( a F c ) )  i^i  ( (,) `  ( b F d ) ) ) )
3938eqeq1d 2412 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( ( (,) `  A )  i^i  ( (,) `  B
) )  =  (/)  <->  (
( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) ) )
4034, 35, 393orbi123d 1253 . . . . . 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 214 . . . . 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 2797 . . . 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 210 . . 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 2795 . 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 188 1  |-  ( ( A  e.  ran  F  /\  B  e.  ran  F )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  \/  ( [,] `  B
)  C_  ( [,] `  A )  \/  (
( (,) `  A
)  i^i  ( (,) `  B ) )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1721   E.wrex 2667    i^i cin 3279    C_ wss 3280   (/)c0 3588   <.cop 3777   class class class wbr 4172    X. cxp 4835   ran crn 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   RRcr 8945   1c1 8947    + caddc 8949    <_ cle 9077    / cdiv 9633   2c2 10005   NN0cn0 10177   ZZcz 10238   (,)cioo 10872   [,]cicc 10875   ^cexp 11337
This theorem is referenced by:  dyadmbl  19445  mblfinlem  26143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-ioo 10876  df-icc 10879  df-seq 11279  df-exp 11338
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