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Theorem omeo 10210
Description: The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
omeo  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  ->  -.  2  ||  ( A  -  B ) )

Proof of Theorem omeo
Dummy variables  a  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 odd2np1 10184 . . . . . 6  |-  ( A  e.  ZZ  ->  ( -.  2  ||  A  <->  E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A ) )
2 2z 8330 . . . . . . 7  |-  2  e.  ZZ
3 divides 10110 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  ||  B  <->  E. b  e.  ZZ  (
b  x.  2 )  =  B ) )
42, 3mpan 408 . . . . . 6  |-  ( B  e.  ZZ  ->  (
2  ||  B  <->  E. b  e.  ZZ  ( b  x.  2 )  =  B ) )
51, 4bi2anan9 548 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  2  ||  B )  <->  ( E. a  e.  ZZ  (
( 2  x.  a
)  +  1 )  =  A  /\  E. b  e.  ZZ  (
b  x.  2 )  =  B ) ) )
6 reeanv 2496 . . . . . 6  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  <-> 
( E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A  /\  E. b  e.  ZZ  ( b  x.  2 )  =  B ) )
7 zsubcl 8343 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  -  b
)  e.  ZZ )
8 zcn 8307 . . . . . . . . . 10  |-  ( a  e.  ZZ  ->  a  e.  CC )
9 zcn 8307 . . . . . . . . . 10  |-  ( b  e.  ZZ  ->  b  e.  CC )
10 2cn 8061 . . . . . . . . . . . . 13  |-  2  e.  CC
11 subdi 7454 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  a  e.  CC  /\  b  e.  CC )  ->  (
2  x.  ( a  -  b ) )  =  ( ( 2  x.  a )  -  ( 2  x.  b
) ) )
1210, 11mp3an1 1230 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
a  -  b ) )  =  ( ( 2  x.  a )  -  ( 2  x.  b ) ) )
1312oveq1d 5555 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  -  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  -  ( 2  x.  b ) )  +  1 ) )
14 mulcl 7066 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  a  e.  CC )  ->  ( 2  x.  a
)  e.  CC )
1510, 14mpan 408 . . . . . . . . . . . 12  |-  ( a  e.  CC  ->  (
2  x.  a )  e.  CC )
16 mulcl 7066 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  e.  CC )
1710, 16mpan 408 . . . . . . . . . . . 12  |-  ( b  e.  CC  ->  (
2  x.  b )  e.  CC )
18 ax-1cn 7035 . . . . . . . . . . . . 13  |-  1  e.  CC
19 addsub 7285 . . . . . . . . . . . . 13  |-  ( ( ( 2  x.  a
)  e.  CC  /\  1  e.  CC  /\  (
2  x.  b )  e.  CC )  -> 
( ( ( 2  x.  a )  +  1 )  -  (
2  x.  b ) )  =  ( ( ( 2  x.  a
)  -  ( 2  x.  b ) )  +  1 ) )
2018, 19mp3an2 1231 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  a
)  e.  CC  /\  ( 2  x.  b
)  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  -  ( 2  x.  b
) )  =  ( ( ( 2  x.  a )  -  (
2  x.  b ) )  +  1 ) )
2115, 17, 20syl2an 277 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  -  (
2  x.  b ) )  =  ( ( ( 2  x.  a
)  -  ( 2  x.  b ) )  +  1 ) )
22 mulcom 7068 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  =  ( b  x.  2 ) )
2310, 22mpan 408 . . . . . . . . . . . . 13  |-  ( b  e.  CC  ->  (
2  x.  b )  =  ( b  x.  2 ) )
2423oveq2d 5556 . . . . . . . . . . . 12  |-  ( b  e.  CC  ->  (
( ( 2  x.  a )  +  1 )  -  ( 2  x.  b ) )  =  ( ( ( 2  x.  a )  +  1 )  -  ( b  x.  2 ) ) )
2524adantl 266 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  -  (
2  x.  b ) )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )
2613, 21, 253eqtr2d 2094 . . . . . . . . . 10  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  -  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )
278, 9, 26syl2an 277 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( 2  x.  ( a  -  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )
28 oveq2 5548 . . . . . . . . . . . 12  |-  ( c  =  ( a  -  b )  ->  (
2  x.  c )  =  ( 2  x.  ( a  -  b
) ) )
2928oveq1d 5555 . . . . . . . . . . 11  |-  ( c  =  ( a  -  b )  ->  (
( 2  x.  c
)  +  1 )  =  ( ( 2  x.  ( a  -  b ) )  +  1 ) )
3029eqeq1d 2064 . . . . . . . . . 10  |-  ( c  =  ( a  -  b )  ->  (
( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) )  <->  ( (
2  x.  ( a  -  b ) )  +  1 )  =  ( ( ( 2  x.  a )  +  1 )  -  (
b  x.  2 ) ) ) )
3130rspcev 2673 . . . . . . . . 9  |-  ( ( ( a  -  b
)  e.  ZZ  /\  ( ( 2  x.  ( a  -  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )
327, 27, 31syl2anc 397 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  -  ( b  x.  2 ) ) )
33 oveq12 5549 . . . . . . . . . 10  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  ( ( ( 2  x.  a )  +  1 )  -  ( b  x.  2 ) )  =  ( A  -  B ) )
3433eqeq2d 2067 . . . . . . . . 9  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  ( ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a )  +  1 )  -  (
b  x.  2 ) )  <->  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
3534rexbidv 2344 . . . . . . . 8  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  ( E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a )  +  1 )  -  ( b  x.  2 ) )  <->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
3632, 35syl5ibcom 148 . . . . . . 7  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
3736rexlimivv 2455 . . . . . 6  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) )
386, 37sylbir 129 . . . . 5  |-  ( ( E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A  /\  E. b  e.  ZZ  (
b  x.  2 )  =  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) )
395, 38syl6bi 156 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  2  ||  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
4039imp 119 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( -.  2  ||  A  /\  2  ||  B ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) )
4140an4s 530 . 2  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) )
42 zsubcl 8343 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B
)  e.  ZZ )
4342ad2ant2r 486 . . 3  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  -> 
( A  -  B
)  e.  ZZ )
44 odd2np1 10184 . . 3  |-  ( ( A  -  B )  e.  ZZ  ->  ( -.  2  ||  ( A  -  B )  <->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
4543, 44syl 14 . 2  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  -> 
( -.  2  ||  ( A  -  B
)  <->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  -  B ) ) )
4641, 45mpbird 160 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 101    <-> wb 102    = wceq 1259    e. wcel 1409   E.wrex 2324   class class class wbr 3792  (class class class)co 5540   CCcc 6945   1c1 6948    + caddc 6950    x. cmul 6952    - cmin 7245   2c2 8040   ZZcz 8302    || cdvds 10108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-xor 1283  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-2 8049  df-n0 8240  df-z 8303  df-dvds 10109
This theorem is referenced by: (None)
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