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Theorem opeo 10441
Description: The sum 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
opeo  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  ->  -.  2  ||  ( A  +  B ) )

Proof of Theorem opeo
Dummy variables  a  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 odd2np1 10417 . . . . . 6  |-  ( A  e.  ZZ  ->  ( -.  2  ||  A  <->  E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A ) )
2 2z 8460 . . . . . . 7  |-  2  e.  ZZ
3 divides 10342 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  ||  B  <->  E. b  e.  ZZ  (
b  x.  2 )  =  B ) )
42, 3mpan 415 . . . . . 6  |-  ( B  e.  ZZ  ->  (
2  ||  B  <->  E. b  e.  ZZ  ( b  x.  2 )  =  B ) )
51, 4bi2anan9 571 . . . . 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 2524 . . . . . 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 zaddcl 8472 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  +  b )  e.  ZZ )
8 zcn 8437 . . . . . . . . . 10  |-  ( a  e.  ZZ  ->  a  e.  CC )
9 zcn 8437 . . . . . . . . . 10  |-  ( b  e.  ZZ  ->  b  e.  CC )
10 2cn 8177 . . . . . . . . . . . . 13  |-  2  e.  CC
11 adddi 7167 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  a  e.  CC  /\  b  e.  CC )  ->  (
2  x.  ( a  +  b ) )  =  ( ( 2  x.  a )  +  ( 2  x.  b
) ) )
1210, 11mp3an1 1256 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
a  +  b ) )  =  ( ( 2  x.  a )  +  ( 2  x.  b ) ) )
1312oveq1d 5558 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  +  b ) )  +  1 )  =  ( ( ( 2  x.  a
)  +  ( 2  x.  b ) )  +  1 ) )
14 mulcl 7162 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  a  e.  CC )  ->  ( 2  x.  a
)  e.  CC )
1510, 14mpan 415 . . . . . . . . . . . 12  |-  ( a  e.  CC  ->  (
2  x.  a )  e.  CC )
16 mulcl 7162 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  e.  CC )
1710, 16mpan 415 . . . . . . . . . . . 12  |-  ( b  e.  CC  ->  (
2  x.  b )  e.  CC )
18 ax-1cn 7131 . . . . . . . . . . . . 13  |-  1  e.  CC
19 add32 7334 . . . . . . . . . . . . 13  |-  ( ( ( 2  x.  a
)  e.  CC  /\  ( 2  x.  b
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2  x.  a )  +  ( 2  x.  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( 2  x.  b ) ) )
2018, 19mp3an3 1258 . . . . . . . . . . . 12  |-  ( ( ( 2  x.  a
)  e.  CC  /\  ( 2  x.  b
)  e.  CC )  ->  ( ( ( 2  x.  a )  +  ( 2  x.  b ) )  +  1 )  =  ( ( ( 2  x.  a )  +  1 )  +  ( 2  x.  b ) ) )
2115, 17, 20syl2an 283 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  ( 2  x.  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( 2  x.  b ) ) )
22 mulcom 7164 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  =  ( b  x.  2 ) )
2310, 22mpan 415 . . . . . . . . . . . . 13  |-  ( b  e.  CC  ->  (
2  x.  b )  =  ( b  x.  2 ) )
2423adantl 271 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  =  ( b  x.  2 ) )
2524oveq2d 5559 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  +  ( 2  x.  b ) )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
2613, 21, 253eqtrd 2118 . . . . . . . . . 10  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  +  b ) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
278, 9, 26syl2an 283 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( 2  x.  ( a  +  b ) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
28 oveq2 5551 . . . . . . . . . . . 12  |-  ( c  =  ( a  +  b )  ->  (
2  x.  c )  =  ( 2  x.  ( a  +  b ) ) )
2928oveq1d 5558 . . . . . . . . . . 11  |-  ( c  =  ( a  +  b )  ->  (
( 2  x.  c
)  +  1 )  =  ( ( 2  x.  ( a  +  b ) )  +  1 ) )
3029eqeq1d 2090 . . . . . . . . . 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 2702 . . . . . . . . 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 403 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
33 oveq12 5552 . . . . . . . . . 10  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  ( ( ( 2  x.  a )  +  1 )  +  ( b  x.  2 ) )  =  ( A  +  B ) )
3433eqeq2d 2093 . . . . . . . . 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 2370 . . . . . . . 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 153 . . . . . . 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 2483 . . . . . 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 133 . . . . 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 161 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  2  ||  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) ) )
4039imp 122 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( -.  2  ||  A  /\  2  ||  B ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) )
4140an4s 553 . 2  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) )
42 zaddcl 8472 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  ZZ )
4342ad2ant2r 493 . . 3  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  -> 
( A  +  B
)  e.  ZZ )
44 odd2np1 10417 . . 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 165 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2350   class class class wbr 3793  (class class class)co 5543   CCcc 7041   1c1 7044    + caddc 7046    x. cmul 7048   2c2 8156   ZZcz 8432    || cdvds 10340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-xor 1308  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-id 4056  df-po 4059  df-iso 4060  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-n0 8356  df-z 8433  df-dvds 10341
This theorem is referenced by: (None)
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