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Theorem opoe 11338
Description: The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
opoe  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  -.  2  ||  B ) )  -> 
2  ||  ( A  +  B ) )

Proof of Theorem opoe
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 odd2np1 11316 . . . . 5  |-  ( A  e.  ZZ  ->  ( -.  2  ||  A  <->  E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A ) )
2 odd2np1 11316 . . . . 5  |-  ( B  e.  ZZ  ->  ( -.  2  ||  B  <->  E. b  e.  ZZ  ( ( 2  x.  b )  +  1 )  =  B ) )
31, 2bi2anan9 574 . . . 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 2550 . . . . 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 8876 . . . . . . . . 9  |-  2  e.  ZZ
6 zaddcl 8888 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  +  b )  e.  ZZ )
76peano2zd 8970 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( a  +  b )  +  1 )  e.  ZZ )
8 dvdsmul1 11261 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( ( a  +  b )  +  1 )  e.  ZZ )  ->  2  ||  (
2  x.  ( ( a  +  b )  +  1 ) ) )
95, 7, 8sylancr 406 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  2  ||  ( 2  x.  ( ( a  +  b )  +  1 ) ) )
10 zcn 8853 . . . . . . . . 9  |-  ( a  e.  ZZ  ->  a  e.  CC )
11 zcn 8853 . . . . . . . . 9  |-  ( b  e.  ZZ  ->  b  e.  CC )
12 addcl 7564 . . . . . . . . . . . . 13  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( a  +  b )  e.  CC )
13 2cn 8591 . . . . . . . . . . . . . 14  |-  2  e.  CC
14 ax-1cn 7535 . . . . . . . . . . . . . 14  |-  1  e.  CC
15 adddi 7571 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  CC  /\  ( a  +  b )  e.  CC  /\  1  e.  CC )  ->  ( 2  x.  (
( a  +  b )  +  1 ) )  =  ( ( 2  x.  ( a  +  b ) )  +  ( 2  x.  1 ) ) )
1613, 14, 15mp3an13 1271 . . . . . . . . . . . . 13  |-  ( ( a  +  b )  e.  CC  ->  (
2  x.  ( ( a  +  b )  +  1 ) )  =  ( ( 2  x.  ( a  +  b ) )  +  ( 2  x.  1 ) ) )
1712, 16syl 14 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
( a  +  b )  +  1 ) )  =  ( ( 2  x.  ( a  +  b ) )  +  ( 2  x.  1 ) ) )
18 adddi 7571 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  CC  /\  a  e.  CC  /\  b  e.  CC )  ->  (
2  x.  ( a  +  b ) )  =  ( ( 2  x.  a )  +  ( 2  x.  b
) ) )
1913, 18mp3an1 1267 . . . . . . . . . . . . 13  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
a  +  b ) )  =  ( ( 2  x.  a )  +  ( 2  x.  b ) ) )
2019oveq1d 5705 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  +  b ) )  +  ( 2  x.  1 ) )  =  ( ( ( 2  x.  a
)  +  ( 2  x.  b ) )  +  ( 2  x.  1 ) ) )
2117, 20eqtrd 2127 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
( a  +  b )  +  1 ) )  =  ( ( ( 2  x.  a
)  +  ( 2  x.  b ) )  +  ( 2  x.  1 ) ) )
22 2t1e2 8667 . . . . . . . . . . . . 13  |-  ( 2  x.  1 )  =  2
23 df-2 8579 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
2422, 23eqtri 2115 . . . . . . . . . . . 12  |-  ( 2  x.  1 )  =  ( 1  +  1 )
2524oveq2i 5701 . . . . . . . . . . 11  |-  ( ( ( 2  x.  a
)  +  ( 2  x.  b ) )  +  ( 2  x.  1 ) )  =  ( ( ( 2  x.  a )  +  ( 2  x.  b
) )  +  ( 1  +  1 ) )
2621, 25syl6eq 2143 . . . . . . . . . 10  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
( a  +  b )  +  1 ) )  =  ( ( ( 2  x.  a
)  +  ( 2  x.  b ) )  +  ( 1  +  1 ) ) )
27 mulcl 7566 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  a  e.  CC )  ->  ( 2  x.  a
)  e.  CC )
2813, 27mpan 416 . . . . . . . . . . 11  |-  ( a  e.  CC  ->  (
2  x.  a )  e.  CC )
29 mulcl 7566 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  e.  CC )
3013, 29mpan 416 . . . . . . . . . . 11  |-  ( b  e.  CC  ->  (
2  x.  b )  e.  CC )
31 add4 7740 . . . . . . . . . . . 12  |-  ( ( ( ( 2  x.  a )  e.  CC  /\  ( 2  x.  b
)  e.  CC )  /\  ( 1  e.  CC  /\  1  e.  CC ) )  -> 
( ( ( 2  x.  a )  +  ( 2  x.  b
) )  +  ( 1  +  1 ) )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( ( 2  x.  b )  +  1 ) ) )
3214, 14, 31mpanr12 431 . . . . . . . . . . 11  |-  ( ( ( 2  x.  a
)  e.  CC  /\  ( 2  x.  b
)  e.  CC )  ->  ( ( ( 2  x.  a )  +  ( 2  x.  b ) )  +  ( 1  +  1 ) )  =  ( ( ( 2  x.  a )  +  1 )  +  ( ( 2  x.  b )  +  1 ) ) )
3328, 30, 32syl2an 284 . . . . . . . . . 10  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  ( 2  x.  b
) )  +  ( 1  +  1 ) )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( ( 2  x.  b )  +  1 ) ) )
3426, 33eqtrd 2127 . . . . . . . . 9  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
( a  +  b )  +  1 ) )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( ( 2  x.  b )  +  1 ) ) )
3510, 11, 34syl2an 284 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( 2  x.  (
( a  +  b )  +  1 ) )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( ( 2  x.  b )  +  1 ) ) )
369, 35breqtrd 3891 . . . . . . 7  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  2  ||  ( ( ( 2  x.  a
)  +  1 )  +  ( ( 2  x.  b )  +  1 ) ) )
37 oveq12 5699 . . . . . . . 8  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  ( ( ( 2  x.  a )  +  1 )  +  ( ( 2  x.  b )  +  1 ) )  =  ( A  +  B ) )
3837breq2d 3879 . . . . . . 7  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  ( 2  ||  ( ( ( 2  x.  a )  +  1 )  +  ( ( 2  x.  b
)  +  1 ) )  <->  2  ||  ( A  +  B )
) )
3936, 38syl5ibcom 154 . . . . . 6  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  2  ||  ( A  +  B
) ) )
4039rexlimivv 2508 . . . . 5  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  2  ||  ( A  +  B )
)
414, 40sylbir 134 . . . 4  |-  ( ( E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A  /\  E. b  e.  ZZ  (
( 2  x.  b
)  +  1 )  =  B )  -> 
2  ||  ( A  +  B ) )
423, 41syl6bi 162 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  -.  2  ||  B )  ->  2  ||  ( A  +  B
) ) )
4342imp 123 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( -.  2  ||  A  /\  -.  2  ||  B ) )  -> 
2  ||  ( A  +  B ) )
4443an4s 556 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 1296    e. wcel 1445   E.wrex 2371   class class class wbr 3867  (class class class)co 5690   CCcc 7445   1c1 7448    + caddc 7450    x. cmul 7452   2c2 8571   ZZcz 8848    || cdvds 11239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560
This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-xor 1319  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-id 4144  df-po 4147  df-iso 4148  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-n0 8772  df-z 8849  df-dvds 11240
This theorem is referenced by: (None)
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