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Theorem dyaddisj 18967
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 18962 . . . 4  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
3 ffn 5405 . . . 4  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  ( ZZ  X.  NN0 ) )
4 ovelrn 6012 . . . . 5  |-  ( F  Fn  ( ZZ  X.  NN0 )  ->  ( A  e.  ran  F  <->  E. a  e.  ZZ  E. c  e. 
NN0  A  =  (
a F c ) ) )
5 ovelrn 6012 . . . . 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 2720 . . 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 2720 . . . 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 9990 . . . . . . . 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 9990 . . . . . . . 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 18966 . . . . . . 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 18966 . . . . . . . . 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 3374 . . . . . . . . . . 11  |-  ( ( (,) `  ( b F d ) )  i^i  ( (,) `  (
a F c ) ) )  =  ( ( (,) `  (
a F c ) )  i^i  ( (,) `  ( b F d ) ) )
2322eqeq1i 2303 . . . . . . . . . 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 8942 . . . . . 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 5545 . . . . . . . 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 5545 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( [,] `  B
)  =  ( [,] `  ( b F d ) ) )
3431, 33sseq12d 3220 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  A )  C_  ( [,] `  B )  <->  ( [,] `  ( a F c ) )  C_  ( [,] `  ( b F d ) ) ) )
3533, 31sseq12d 3220 . . . . . . 7  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( [,] `  B )  C_  ( [,] `  A )  <->  ( [,] `  ( b F d ) )  C_  ( [,] `  ( a F c ) ) ) )
3630fveq2d 5545 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( (,) `  A
)  =  ( (,) `  ( a F c ) ) )
3732fveq2d 5545 . . . . . . . . 9  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( (,) `  B
)  =  ( (,) `  ( b F d ) ) )
3836, 37ineq12d 3384 . . . . . . . 8  |-  ( ( A  =  ( a F c )  /\  B  =  ( b F d ) )  ->  ( ( (,) `  A )  i^i  ( (,) `  B ) )  =  ( ( (,) `  ( a F c ) )  i^i  ( (,) `  ( b F d ) ) ) )
3938eqeq1d 2304 . . . . . . 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 2687 . . . 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 2685 . 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 1632    e. wcel 1696   E.wrex 2557    i^i cin 3164    C_ wss 3165   (/)c0 3468   <.cop 3656   class class class wbr 4039    X. cxp 4703   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   RRcr 8752   1c1 8754    + caddc 8756    <_ cle 8884    / cdiv 9439   2c2 9811   NN0cn0 9981   ZZcz 10040   (,)cioo 10672   [,]cicc 10675   ^cexp 11120
This theorem is referenced by:  dyadmbl  18971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-ioo 10676  df-icc 10679  df-seq 11063  df-exp 11121
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