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Theorem opoe 11914
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 11892 . . . . 5  |-  ( A  e.  ZZ  ->  ( -.  2  ||  A  <->  E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A ) )
2 odd2np1 11892 . . . . 5  |-  ( B  e.  ZZ  ->  ( -.  2  ||  B  <->  E. b  e.  ZZ  ( ( 2  x.  b )  +  1 )  =  B ) )
31, 2bi2anan9 606 . . . 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 2657 . . . . 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 9295 . . . . . . . . 9  |-  2  e.  ZZ
6 zaddcl 9307 . . . . . . . . . 10  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  +  b )  e.  ZZ )
76peano2zd 9392 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( a  +  b )  +  1 )  e.  ZZ )
8 dvdsmul1 11834 . . . . . . . . 9  |-  ( ( 2  e.  ZZ  /\  ( ( a  +  b )  +  1 )  e.  ZZ )  ->  2  ||  (
2  x.  ( ( a  +  b )  +  1 ) ) )
95, 7, 8sylancr 414 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  2  ||  ( 2  x.  ( ( a  +  b )  +  1 ) ) )
10 zcn 9272 . . . . . . . . 9  |-  ( a  e.  ZZ  ->  a  e.  CC )
11 zcn 9272 . . . . . . . . 9  |-  ( b  e.  ZZ  ->  b  e.  CC )
12 addcl 7950 . . . . . . . . . . . . 13  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( a  +  b )  e.  CC )
13 2cn 9004 . . . . . . . . . . . . . 14  |-  2  e.  CC
14 ax-1cn 7918 . . . . . . . . . . . . . 14  |-  1  e.  CC
15 adddi 7957 . . . . . . . . . . . . . 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 1338 . . . . . . . . . . . . 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 7957 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  CC  /\  a  e.  CC  /\  b  e.  CC )  ->  (
2  x.  ( a  +  b ) )  =  ( ( 2  x.  a )  +  ( 2  x.  b
) ) )
1913, 18mp3an1 1334 . . . . . . . . . . . . 13  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
a  +  b ) )  =  ( ( 2  x.  a )  +  ( 2  x.  b ) ) )
2019oveq1d 5903 . . . . . . . . . . . 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 2220 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
( a  +  b )  +  1 ) )  =  ( ( ( 2  x.  a
)  +  ( 2  x.  b ) )  +  ( 2  x.  1 ) ) )
22 2t1e2 9086 . . . . . . . . . . . . 13  |-  ( 2  x.  1 )  =  2
23 df-2 8992 . . . . . . . . . . . . 13  |-  2  =  ( 1  +  1 )
2422, 23eqtri 2208 . . . . . . . . . . . 12  |-  ( 2  x.  1 )  =  ( 1  +  1 )
2524oveq2i 5899 . . . . . . . . . . 11  |-  ( ( ( 2  x.  a
)  +  ( 2  x.  b ) )  +  ( 2  x.  1 ) )  =  ( ( ( 2  x.  a )  +  ( 2  x.  b
) )  +  ( 1  +  1 ) )
2621, 25eqtrdi 2236 . . . . . . . . . 10  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
( a  +  b )  +  1 ) )  =  ( ( ( 2  x.  a
)  +  ( 2  x.  b ) )  +  ( 1  +  1 ) ) )
27 mulcl 7952 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  a  e.  CC )  ->  ( 2  x.  a
)  e.  CC )
2813, 27mpan 424 . . . . . . . . . . 11  |-  ( a  e.  CC  ->  (
2  x.  a )  e.  CC )
29 mulcl 7952 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  e.  CC )
3013, 29mpan 424 . . . . . . . . . . 11  |-  ( b  e.  CC  ->  (
2  x.  b )  e.  CC )
31 add4 8132 . . . . . . . . . . . 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 439 . . . . . . . . . . 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 289 . . . . . . . . . 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 2220 . . . . . . . . 9  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
( a  +  b )  +  1 ) )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( ( 2  x.  b )  +  1 ) ) )
3510, 11, 34syl2an 289 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( 2  x.  (
( a  +  b )  +  1 ) )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( ( 2  x.  b )  +  1 ) ) )
369, 35breqtrd 4041 . . . . . . 7  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  2  ||  ( ( ( 2  x.  a
)  +  1 )  +  ( ( 2  x.  b )  +  1 ) ) )
37 oveq12 5897 . . . . . . . 8  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  ( ( ( 2  x.  a )  +  1 )  +  ( ( 2  x.  b )  +  1 ) )  =  ( A  +  B ) )
3837breq2d 4027 . . . . . . 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 155 . . . . . 6  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  2  ||  ( A  +  B
) ) )
4039rexlimivv 2610 . . . . 5  |-  ( E. a  e.  ZZ  E. b  e.  ZZ  (
( ( 2  x.  a )  +  1 )  =  A  /\  ( ( 2  x.  b )  +  1 )  =  B )  ->  2  ||  ( A  +  B )
)
414, 40sylbir 135 . . . 4  |-  ( ( E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A  /\  E. b  e.  ZZ  (
( 2  x.  b
)  +  1 )  =  B )  -> 
2  ||  ( A  +  B ) )
423, 41biimtrdi 163 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  -.  2  ||  B )  ->  2  ||  ( A  +  B
) ) )
4342imp 124 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( -.  2  ||  A  /\  -.  2  ||  B ) )  -> 
2  ||  ( A  +  B ) )
4443an4s 588 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 104    = wceq 1363    e. wcel 2158   E.wrex 2466   class class class wbr 4015  (class class class)co 5888   CCcc 7823   1c1 7826    + caddc 7828    x. cmul 7830   2c2 8984   ZZcz 9267    || cdvds 11808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-xor 1386  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-n0 9191  df-z 9268  df-dvds 11809
This theorem is referenced by:  pythagtriplem11  12288
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