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Theorem omoe 11582
Description: The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
omoe  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  -.  2  ||  B ) )  -> 
2  ||  ( A  -  B ) )

Proof of Theorem omoe
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 odd2np1 11559 . . . . 5  |-  ( A  e.  ZZ  ->  ( -.  2  ||  A  <->  E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A ) )
2 odd2np1 11559 . . . . 5  |-  ( B  e.  ZZ  ->  ( -.  2  ||  B  <->  E. b  e.  ZZ  ( ( 2  x.  b )  +  1 )  =  B ) )
31, 2bi2anan9 595 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  -.  2  ||  B )  <->  ( E. a  e.  ZZ  (
( 2  x.  a
)  +  1 )  =  A  /\  E. b  e.  ZZ  (
( 2  x.  b
)  +  1 )  =  B ) ) )
4 reeanv 2598 . . . . 5  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  <-> 
( E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A  /\  E. b  e.  ZZ  ( ( 2  x.  b )  +  1 )  =  B ) )
5 2z 9075 . . . . . . . . 9  |-  2  e.  ZZ
6 zsubcl 9088 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  -  b
)  e.  ZZ )
7 dvdsmul1 11504 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( a  -  b
)  e.  ZZ )  ->  2  ||  (
2  x.  ( a  -  b ) ) )
85, 6, 7sylancr 410 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  2  ||  ( 2  x.  ( a  -  b ) ) )
9 zcn 9052 . . . . . . . . 9  |-  ( a  e.  ZZ  ->  a  e.  CC )
10 zcn 9052 . . . . . . . . 9  |-  ( b  e.  ZZ  ->  b  e.  CC )
11 2cn 8784 . . . . . . . . . . . 12  |-  2  e.  CC
12 mulcl 7740 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  a  e.  CC )  ->  ( 2  x.  a
)  e.  CC )
1311, 12mpan 420 . . . . . . . . . . 11  |-  ( a  e.  CC  ->  (
2  x.  a )  e.  CC )
14 mulcl 7740 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  e.  CC )
1511, 14mpan 420 . . . . . . . . . . 11  |-  ( b  e.  CC  ->  (
2  x.  b )  e.  CC )
16 ax-1cn 7706 . . . . . . . . . . . 12  |-  1  e.  CC
17 pnpcan2 7995 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  a
)  e.  CC  /\  ( 2  x.  b
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  -  (
( 2  x.  b
)  +  1 ) )  =  ( ( 2  x.  a )  -  ( 2  x.  b ) ) )
1816, 17mp3an3 1304 . . . . . . . . . . 11  |-  ( ( ( 2  x.  a
)  e.  CC  /\  ( 2  x.  b
)  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  -  ( ( 2  x.  b )  +  1 ) )  =  ( ( 2  x.  a
)  -  ( 2  x.  b ) ) )
1913, 15, 18syl2an 287 . . . . . . . . . 10  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  -  (
( 2  x.  b
)  +  1 ) )  =  ( ( 2  x.  a )  -  ( 2  x.  b ) ) )
20 subdi 8140 . . . . . . . . . . 11  |-  ( ( 2  e.  CC  /\  a  e.  CC  /\  b  e.  CC )  ->  (
2  x.  ( a  -  b ) )  =  ( ( 2  x.  a )  -  ( 2  x.  b
) ) )
2111, 20mp3an1 1302 . . . . . . . . . 10  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
a  -  b ) )  =  ( ( 2  x.  a )  -  ( 2  x.  b ) ) )
2219, 21eqtr4d 2173 . . . . . . . . 9  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  -  (
( 2  x.  b
)  +  1 ) )  =  ( 2  x.  ( a  -  b ) ) )
239, 10, 22syl2an 287 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( ( 2  x.  a )  +  1 )  -  (
( 2  x.  b
)  +  1 ) )  =  ( 2  x.  ( a  -  b ) ) )
248, 23breqtrrd 3951 . . . . . . 7  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  2  ||  ( ( ( 2  x.  a
)  +  1 )  -  ( ( 2  x.  b )  +  1 ) ) )
25 oveq12 5776 . . . . . . . 8  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  ( ( ( 2  x.  a )  +  1 )  -  ( ( 2  x.  b )  +  1 ) )  =  ( A  -  B ) )
2625breq2d 3936 . . . . . . 7  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  ( 2  ||  ( ( ( 2  x.  a )  +  1 )  -  (
( 2  x.  b
)  +  1 ) )  <->  2  ||  ( A  -  B )
) )
2724, 26syl5ibcom 154 . . . . . 6  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  2  ||  ( A  -  B
) ) )
2827rexlimivv 2553 . . . . 5  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  2  ||  ( A  -  B )
)
294, 28sylbir 134 . . . 4  |-  ( ( E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A  /\  E. b  e.  ZZ  (
( 2  x.  b
)  +  1 )  =  B )  -> 
2  ||  ( A  -  B ) )
303, 29syl6bi 162 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  -.  2  ||  B )  ->  2  ||  ( A  -  B
) ) )
3130imp 123 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( -.  2  ||  A  /\  -.  2  ||  B ) )  -> 
2  ||  ( A  -  B ) )
3231an4s 577 1  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  -.  2  ||  B ) )  -> 
2  ||  ( A  -  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   E.wrex 2415   class class class wbr 3924  (class class class)co 5767   CCcc 7611   1c1 7614    + caddc 7616    x. cmul 7618    - cmin 7926   2c2 8764   ZZcz 9047    || cdvds 11482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-xor 1354  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-n0 8971  df-z 9048  df-dvds 11483
This theorem is referenced by: (None)
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