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Theorem opeo 14213
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 13922 . . . . . 6  |-  ( A  e.  ZZ  ->  ( -.  2  ||  A  <->  E. a  e.  ZZ  ( ( 2  x.  a )  +  1 )  =  A ) )
2 2z 10908 . . . . . . 7  |-  2  e.  ZZ
3 divides 13866 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  B  e.  ZZ )  ->  ( 2  ||  B  <->  E. b  e.  ZZ  (
b  x.  2 )  =  B ) )
42, 3mpan 670 . . . . . 6  |-  ( B  e.  ZZ  ->  (
2  ||  B  <->  E. b  e.  ZZ  ( b  x.  2 )  =  B ) )
51, 4bi2anan9 871 . . . . 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 3034 . . . . . 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 10915 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( a  +  b )  e.  ZZ )
8 zcn 10881 . . . . . . . . . 10  |-  ( a  e.  ZZ  ->  a  e.  CC )
9 zcn 10881 . . . . . . . . . 10  |-  ( b  e.  ZZ  ->  b  e.  CC )
10 2cn 10618 . . . . . . . . . . . . 13  |-  2  e.  CC
11 adddi 9593 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  a  e.  CC  /\  b  e.  CC )  ->  (
2  x.  ( a  +  b ) )  =  ( ( 2  x.  a )  +  ( 2  x.  b
) ) )
1210, 11mp3an1 1311 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  (
a  +  b ) )  =  ( ( 2  x.  a )  +  ( 2  x.  b ) ) )
1312oveq1d 6310 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  +  b ) )  +  1 )  =  ( ( ( 2  x.  a
)  +  ( 2  x.  b ) )  +  1 ) )
14 mulcl 9588 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  a  e.  CC )  ->  ( 2  x.  a
)  e.  CC )
1510, 14mpan 670 . . . . . . . . . . . 12  |-  ( a  e.  CC  ->  (
2  x.  a )  e.  CC )
16 mulcl 9588 . . . . . . . . . . . . 13  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  e.  CC )
1710, 16mpan 670 . . . . . . . . . . . 12  |-  ( b  e.  CC  ->  (
2  x.  b )  e.  CC )
18 ax-1cn 9562 . . . . . . . . . . . . 13  |-  1  e.  CC
19 add32 9805 . . . . . . . . . . . . 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 1313 . . . . . . . . . . . 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 477 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  ( 2  x.  b
) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( 2  x.  b ) ) )
22 mulcom 9590 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  =  ( b  x.  2 ) )
2310, 22mpan 670 . . . . . . . . . . . . 13  |-  ( b  e.  CC  ->  (
2  x.  b )  =  ( b  x.  2 ) )
2423adantl 466 . . . . . . . . . . . 12  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( 2  x.  b
)  =  ( b  x.  2 ) )
2524oveq2d 6311 . . . . . . . . . . 11  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( ( 2  x.  a )  +  1 )  +  ( 2  x.  b ) )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
2613, 21, 253eqtrd 2512 . . . . . . . . . 10  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( ( 2  x.  ( a  +  b ) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
278, 9, 26syl2an 477 . . . . . . . . 9  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  ( ( 2  x.  ( a  +  b ) )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
28 oveq2 6303 . . . . . . . . . . . 12  |-  ( c  =  ( a  +  b )  ->  (
2  x.  c )  =  ( 2  x.  ( a  +  b ) ) )
2928oveq1d 6310 . . . . . . . . . . 11  |-  ( c  =  ( a  +  b )  ->  (
( 2  x.  c
)  +  1 )  =  ( ( 2  x.  ( a  +  b ) )  +  1 ) )
3029eqeq1d 2469 . . . . . . . . . 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 3219 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ( a  e.  ZZ  /\  b  e.  ZZ )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( ( ( 2  x.  a
)  +  1 )  +  ( b  x.  2 ) ) )
33 oveq12 6304 . . . . . . . . . 10  |-  ( ( ( ( 2  x.  a )  +  1 )  =  A  /\  ( b  x.  2 )  =  B )  ->  ( ( ( 2  x.  a )  +  1 )  +  ( b  x.  2 ) )  =  ( A  +  B ) )
3433eqeq2d 2481 . . . . . . . . 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 2978 . . . . . . . 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 220 . . . . . . 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 2964 . . . . . 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 213 . . . . 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 228 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( -.  2  ||  A  /\  2  ||  B )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) ) )
4039imp 429 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  ( -.  2  ||  A  /\  2  ||  B ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) )
4140an4s 824 . 2  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  ->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) )
42 zaddcl 10915 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  ZZ )
4342ad2ant2r 746 . . 3  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  -> 
( A  +  B
)  e.  ZZ )
44 odd2np1 13922 . . 3  |-  ( ( A  +  B )  e.  ZZ  ->  ( -.  2  ||  ( A  +  B )  <->  E. c  e.  ZZ  ( ( 2  x.  c )  +  1 )  =  ( A  +  B ) ) )
4543, 44syl 16 . 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 232 1  |-  ( ( ( A  e.  ZZ  /\ 
-.  2  ||  A
)  /\  ( B  e.  ZZ  /\  2  ||  B ) )  ->  -.  2  ||  ( A  +  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   class class class wbr 4453  (class class class)co 6295   CCcc 9502   1c1 9505    + caddc 9507    x. cmul 9509   2c2 10597   ZZcz 10876    || cdivides 13864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-dvds 13865
This theorem is referenced by: (None)
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