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Theorem dyaddisj 18951
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 18946 . . . 4  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
3 ffn 5389 . . . 4  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  ( ZZ  X.  NN0 ) )
4 ovelrn 5996 . . . . 5  |-  ( F  Fn  ( ZZ  X.  NN0 )  ->  ( A  e.  ran  F  <->  E. a  e.  ZZ  E. c  e. 
NN0  A  =  (
a F c ) ) )
5 ovelrn 5996 . . . . 5  |-  ( F  Fn  ( ZZ  X.  NN0 )  ->  ( B  e.  ran  F  <->  E. b  e.  ZZ  E. d  e. 
NN0  B  =  (
b F d ) ) )
64, 5anbi12d 691 . . . 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 9 . . 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 2707 . . 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 243 . 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 2707 . . . 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 9974 . . . . . . . 8  |-  ( c  e.  NN0  ->  c  e.  RR )
1211ad2antrl 708 . . . . . . 7  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  ->  c  e.  RR )
13 nn0re 9974 . . . . . . . 8  |-  ( d  e.  NN0  ->  d  e.  RR )
1413ad2antll 709 . . . . . . 7  |-  ( ( ( a  e.  ZZ  /\  b  e.  ZZ )  /\  ( c  e. 
NN0  /\  d  e.  NN0 ) )  ->  d  e.  RR )
151dyaddisjlem 18950 . . . . . . 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 437 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  <->  ( b  e.  ZZ  /\  a  e.  ZZ )
)
17 ancom 437 . . . . . . . . . 10  |-  ( ( c  e.  NN0  /\  d  e.  NN0 )  <->  ( d  e.  NN0  /\  c  e. 
NN0 ) )
1816, 17anbi12i 678 . . . . . . . . 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 18950 . . . . . . . . 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 458 . . . . . . . 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 376 . . . . . . . . . 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 3361 . . . . . . . . . . 11  |-  ( ( (,) `  ( b F d ) )  i^i  ( (,) `  (
a F c ) ) )  =  ( ( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )
2322eqeq1i 2290 . . . . . . . . . 10  |-  ( ( ( (,) `  (
b F d ) )  i^i  ( (,) `  ( a F c ) ) )  =  (/) 
<->  ( ( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) )
2421, 23orbi12i 507 . . . . . . . . 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 935 . . . . . . . . 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 935 . . . . . . . . 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 268 . . . . . . . 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 188 . . . . . . 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 8926 . . . . . 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 443 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  A  =  ( a F c ) )
3130fveq2d 5529 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( [,] `  A
)  =  ( [,] `  ( a F c ) ) )
32 simpr 447 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  B  =  ( b F d ) )
3332fveq2d 5529 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( [,] `  B
)  =  ( [,] `  ( b F d ) ) )
3431, 33sseq12d 3207 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  <->  ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) ) ) )
3533, 31sseq12d 3207 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  B )  C_  ( [,] `  A )  <->  ( [,] `  ( b F d ) )  C_  ( [,] `  ( a F c ) ) ) )
3630fveq2d 5529 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( (,) `  A
)  =  ( (,) `  ( a F c ) ) )
3732fveq2d 5529 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( (,) `  B
)  =  ( (,) `  ( b F d ) ) )
3836, 37ineq12d 3371 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( (,) `  A )  i^i  ( (,) `  B ) )  =  ( ( (,) `  ( a F c ) )  i^i  ( (,) `  ( b F d ) ) ) )
3938eqeq1d 2291 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( ( (,) `  A )  i^i  ( (,) `  B
) )  =  (/)  <->  (
( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )  =  (/) ) )
4034, 35, 393orbi123d 1251 . . . . . 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 213 . . . . 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 2674 . . . 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 209 . . 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 2672 . 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 187 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 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   <.cop 3643   class class class wbr 4023    X. cxp 4687   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   RRcr 8736   1c1 8738    + caddc 8740    <_ cle 8868    / cdiv 9423   2c2 9795   NN0cn0 9965   ZZcz 10024   (,)cioo 10656   [,]cicc 10659   ^cexp 11104
This theorem is referenced by:  dyadmbl  18955
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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
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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-n0 9966  df-z 10025  df-uz 10231  df-ioo 10660  df-icc 10663  df-seq 11047  df-exp 11105
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