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Theorem dyaddisj 22610
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 22605 . . . 4  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
3 ffn 5755 . . . 4  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  ( ZZ  X.  NN0 ) )
4 ovelrn 6477 . . . . 5  |-  ( F  Fn  ( ZZ  X.  NN0 )  ->  ( A  e.  ran  F  <->  E. a  e.  ZZ  E. c  e. 
NN0  A  =  (
a F c ) ) )
5 ovelrn 6477 . . . . 5  |-  ( F  Fn  ( ZZ  X.  NN0 )  ->  ( B  e.  ran  F  <->  E. b  e.  ZZ  E. d  e. 
NN0  B  =  (
b F d ) ) )
64, 5anbi12d 722 . . . 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 2970 . . 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 260 . 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 2970 . . . 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 10912 . . . . . . . 8  |-  ( c  e.  NN0  ->  c  e.  RR )
1211ad2antrl 739 . . . . . . 7  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  ->  c  e.  RR )
13 nn0re 10912 . . . . . . . 8  |-  ( d  e.  NN0  ->  d  e.  RR )
1413ad2antll 740 . . . . . . 7  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  ->  d  e.  RR )
151dyaddisjlem 22609 . . . . . . 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 456 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  <->  ( b  e.  ZZ  /\  a  e.  ZZ )
)
17 ancom 456 . . . . . . . . . 10  |-  ( ( c  e.  NN0  /\  d  e.  NN0 )  <->  ( d  e.  NN0  /\  c  e. 
NN0 ) )
1816, 17anbi12i 708 . . . . . . . . 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 22609 . . . . . . . . 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 479 . . . . . . . 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 393 . . . . . . . . . 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 3637 . . . . . . . . . . 11  |-  ( ( (,) `  ( b F d ) )  i^i  ( (,) `  (
a F c ) ) )  =  ( ( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )
2322eqeq1i 2467 . . . . . . . . . 10  |-  ( ( ( (,) `  (
b F d ) )  i^i  ( (,) `  ( a F c ) ) )  =  (/) 
<->  ( ( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) )
2421, 23orbi12i 528 . . . . . . . . 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 992 . . . . . . . . 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 992 . . . . . . . . 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 285 . . . . . . . 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 201 . . . . . . 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 9771 . . . . . 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 463 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  A  =  ( a F c ) )
3130fveq2d 5896 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( [,] `  A
)  =  ( [,] `  ( a F c ) ) )
32 simpr 467 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  B  =  ( b F d ) )
3332fveq2d 5896 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( [,] `  B
)  =  ( [,] `  ( b F d ) ) )
3431, 33sseq12d 3473 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  <->  ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) ) ) )
3533, 31sseq12d 3473 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  B )  C_  ( [,] `  A )  <->  ( [,] `  ( b F d ) )  C_  ( [,] `  ( a F c ) ) ) )
3630fveq2d 5896 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( (,) `  A
)  =  ( (,) `  ( a F c ) ) )
3732fveq2d 5896 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( (,) `  B
)  =  ( (,) `  ( b F d ) ) )
3836, 37ineq12d 3647 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( (,) `  A )  i^i  ( (,) `  B ) )  =  ( ( (,) `  ( a F c ) )  i^i  ( (,) `  ( b F d ) ) ) )
3938eqeq1d 2464 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( ( (,) `  A )  i^i  ( (,) `  B
) )  =  (/)  <->  (
( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) ) )
4034, 35, 393orbi123d 1347 . . . . . 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 230 . . . . 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 2898 . . . 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 226 . . 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 2896 . 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 200 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 189    \/ wo 374    /\ wa 375    \/ w3o 990    = wceq 1455    e. wcel 1898   E.wrex 2750    i^i cin 3415    C_ wss 3416   (/)c0 3743   <.cop 3986   class class class wbr 4418    X. cxp 4854   ran crn 4857    Fn wfn 5600   -->wf 5601   ` cfv 5605  (class class class)co 6320    |-> cmpt2 6322   RRcr 9569   1c1 9571    + caddc 9573    <_ cle 9707    / cdiv 10302   2c2 10692   NN0cn0 10903   ZZcz 10971   (,)cioo 11669   [,]cicc 11672   ^cexp 12310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-n0 10904  df-z 10972  df-uz 11194  df-ioo 11673  df-icc 11676  df-seq 12252  df-exp 12311
This theorem is referenced by:  dyadmbl  22614  mblfinlem2  32024
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