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Theorem opeo 12882
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 12603 . . . . . 6  |-  ( A  e.  ZZ  ->  ( -.  2  ||  A  <->  E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A ) )
2 2z 10070 . . . . . . 7  |-  2  e.  ZZ
3 divides 12549 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  ||  B  <->  E. b  e.  ZZ  (
b  x.  2 )  =  B ) )
42, 3mpan 651 . . . . . 6  |-  ( B  e.  ZZ  ->  (
2  ||  B  <->  E. b  e.  ZZ  ( b  x.  2 )  =  B ) )
51, 4bi2anan9 843 . . . . 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 2720 . . . . . 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 10075 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  +  b )  e.  ZZ )
8 zcn 10045 . . . . . . . . . 10  |-  ( a  e.  ZZ  ->  a  e.  CC )
9 zcn 10045 . . . . . . . . . 10  |-  ( b  e.  ZZ  ->  b  e.  CC )
10 2cn 9832 . . . . . . . . . . . . 13  |-  2  e.  CC
11 adddi 8842 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  a  e.  CC  /\  b  e.  CC )  ->  (
2  x.  ( a  +  b ) )  =  ( ( 2  x.  a )  +  ( 2  x.  b
) ) )
1210, 11mp3an1 1264 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
a  +  b ) )  =  ( ( 2  x.  a )  +  ( 2  x.  b ) ) )
1312oveq1d 5889 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  +  b ) )  +  1 )  =  ( ( ( 2  x.  a
)  +  ( 2  x.  b ) )  +  1 ) )
14 mulcl 8837 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  a  e.  CC )  ->  ( 2  x.  a
)  e.  CC )
1510, 14mpan 651 . . . . . . . . . . . 12  |-  ( a  e.  CC  ->  (
2  x.  a )  e.  CC )
16 mulcl 8837 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  e.  CC )
1710, 16mpan 651 . . . . . . . . . . . 12  |-  ( b  e.  CC  ->  (
2  x.  b )  e.  CC )
18 ax-1cn 8811 . . . . . . . . . . . . 13  |-  1  e.  CC
19 add32 9042 . . . . . . . . . . . . 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 1266 . . . . . . . . . . . 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 463 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  ( 2  x.  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( 2  x.  b ) ) )
22 mulcom 8839 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  =  ( b  x.  2 ) )
2310, 22mpan 651 . . . . . . . . . . . . 13  |-  ( b  e.  CC  ->  (
2  x.  b )  =  ( b  x.  2 ) )
2423adantl 452 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  =  ( b  x.  2 ) )
2524oveq2d 5890 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  +  ( 2  x.  b ) )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
2613, 21, 253eqtrd 2332 . . . . . . . . . 10  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  +  b ) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
278, 9, 26syl2an 463 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( 2  x.  ( a  +  b ) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
28 oveq2 5882 . . . . . . . . . . . 12  |-  ( c  =  ( a  +  b )  ->  (
2  x.  c )  =  ( 2  x.  ( a  +  b ) ) )
2928oveq1d 5889 . . . . . . . . . . 11  |-  ( c  =  ( a  +  b )  ->  (
( 2  x.  c
)  +  1 )  =  ( ( 2  x.  ( a  +  b ) )  +  1 ) )
3029eqeq1d 2304 . . . . . . . . . 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 2897 . . . . . . . . 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 642 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
33 oveq12 5883 . . . . . . . . . 10  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  ( ( ( 2  x.  a )  +  1 )  +  ( b  x.  2 ) )  =  ( A  +  B ) )
3433eqeq2d 2307 . . . . . . . . 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 2577 . . . . . . . 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 211 . . . . . . 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 2685 . . . . . 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 204 . . . . 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 219 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  2  ||  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) ) )
4039imp 418 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( -.  2  ||  A  /\  2  ||  B ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) )
4140an4s 799 . 2  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) )
42 zaddcl 10075 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  ZZ )
4342ad2ant2r 727 . . 3  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  -> 
( A  +  B
)  e.  ZZ )
44 odd2np1 12603 . . 3  |-  ( ( A  +  B )  e.  ZZ  ->  ( -.  2  ||  ( A  +  B )  <->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) ) )
4543, 44syl 15 . 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 223 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    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    x. cmul 8758   2c2 9811   ZZcz 10040    || cdivides 12547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-dvds 12548
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