Theorem opeo 11902

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.

Step	Hyp	Ref	Expression
1		odd2np1 11878	. . . . . 6 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		2z 9281	. . . . . . 7 |- 2 e. ZZ
3		divides 11796	. . . . . . 7 |- ( ( 2 e. ZZ /\ B e. ZZ ) -> ( 2 || B <-> E. b e. ZZ ( b x. 2 ) = B ) )
4	2, 3	mpan 424	. . . . . 6 |- ( B e. ZZ -> ( 2 || B <-> E. b e. ZZ ( b x. 2 ) = B ) )
5	1, 4	bi2anan9 606	. . . . 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 2647	. . . . . 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 9293	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a + b ) e. ZZ )
8		zcn 9258	. . . . . . . . . 10 |- ( a e. ZZ -> a e. CC )
9		zcn 9258	. . . . . . . . . 10 |- ( b e. ZZ -> b e. CC )
10		2cn 8990	. . . . . . . . . . . . 13 |- 2 e. CC
11		adddi 7943	. . . . . . . . . . . . 13 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a + b ) ) = ( ( 2 x. a ) + ( 2 x. b ) ) )
12	10, 11	mp3an1 1324	. . . . . . . . . . . 12 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a + b ) ) = ( ( 2 x. a ) + ( 2 x. b ) ) )
13	12	oveq1d 5890	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( ( 2 x. ( a + b ) ) + 1 ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + 1 ) )
14		mulcl 7938	. . . . . . . . . . . . 13 |- ( ( 2 e. CC /\ a e. CC ) -> ( 2 x. a ) e. CC )
15	10, 14	mpan 424	. . . . . . . . . . . 12 |- ( a e. CC -> ( 2 x. a ) e. CC )
16		mulcl 7938	. . . . . . . . . . . . 13 |- ( ( 2 e. CC /\ b e. CC ) -> ( 2 x. b ) e. CC )
17	10, 16	mpan 424	. . . . . . . . . . . 12 |- ( b e. CC -> ( 2 x. b ) e. CC )
18		ax-1cn 7904	. . . . . . . . . . . . 13 |- 1 e. CC
19		add32 8116	. . . . . . . . . . . . 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 ) ) )
20	18, 19	mp3an3 1326	. . . . . . . . . . . 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 ) ) )
21	15, 17, 20	syl2an 289	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( ( ( 2 x. a ) + ( 2 x. b ) ) + 1 ) = ( ( ( 2 x. a ) + 1 ) + ( 2 x. b ) ) )
22		mulcom 7940	. . . . . . . . . . . . . 14 |- ( ( 2 e. CC /\ b e. CC ) -> ( 2 x. b ) = ( b x. 2 ) )
23	10, 22	mpan 424	. . . . . . . . . . . . 13 |- ( b e. CC -> ( 2 x. b ) = ( b x. 2 ) )
24	23	adantl 277	. . . . . . . . . . . 12 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. b ) = ( b x. 2 ) )
25	24	oveq2d 5891	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( ( ( 2 x. a ) + 1 ) + ( 2 x. b ) ) = ( ( ( 2 x. a ) + 1 ) + ( b x. 2 ) ) )
26	13, 21, 25	3eqtrd 2214	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( ( 2 x. ( a + b ) ) + 1 ) = ( ( ( 2 x. a ) + 1 ) + ( b x. 2 ) ) )
27	8, 9, 26	syl2an 289	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( 2 x. ( a + b ) ) + 1 ) = ( ( ( 2 x. a ) + 1 ) + ( b x. 2 ) ) )
28		oveq2 5883	. . . . . . . . . . . 12 |- ( c = ( a + b ) -> ( 2 x. c ) = ( 2 x. ( a + b ) ) )
29	28	oveq1d 5890	. . . . . . . . . . 11 |- ( c = ( a + b ) -> ( ( 2 x. c ) + 1 ) = ( ( 2 x. ( a + b ) ) + 1 ) )
30	29	eqeq1d 2186	. . . . . . . . . 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 ) ) ) )
31	30	rspcev 2842	. . . . . . . . 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 ) ) )
32	7, 27, 31	syl2anc 411	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ ) -> E. c e. ZZ ( ( 2 x. c ) + 1 ) = ( ( ( 2 x. a ) + 1 ) + ( b x. 2 ) ) )
33		oveq12 5884	. . . . . . . . . 10 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( b x. 2 ) = B ) -> ( ( ( 2 x. a ) + 1 ) + ( b x. 2 ) ) = ( A + B ) )
34	33	eqeq2d 2189	. . . . . . . . 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 ) ) )
35	34	rexbidv 2478	. . . . . . . 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 ) ) )
36	32, 35	syl5ibcom 155	. . . . . . 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 ) ) )
37	36	rexlimivv 2600	. . . . . 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 ) )
38	6, 37	sylbir 135	. . . . 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 ) )
39	5, 38	syl6bi 163	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( -. 2 || A /\ 2 || B ) -> E. c e. ZZ ( ( 2 x. c ) + 1 ) = ( A + B ) ) )
40	39	imp 124	. . 3 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( -. 2 || A /\ 2 || B ) ) -> E. c e. ZZ ( ( 2 x. c ) + 1 ) = ( A + B ) )
41	40	an4s 588	. 2 |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) -> E. c e. ZZ ( ( 2 x. c ) + 1 ) = ( A + B ) )
42		zaddcl 9293	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A + B ) e. ZZ )
43	42	ad2ant2r 509	. . 3 |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) -> ( A + B ) e. ZZ )
44		odd2np1 11878	. . 3 |- ( ( A + B ) e. ZZ -> ( -. 2 || ( A + B ) <-> E. c e. ZZ ( ( 2 x. c ) + 1 ) = ( A + B ) ) )
45	43, 44	syl 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 ) ) )
46	41, 45	mpbird 167	1 |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) -> -. 2 || ( A + B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4004  (class class class)co 5875   CCcc 7809   1c1 7812   + caddc 7814   x. cmul 7816   2c2 8970   ZZcz 9253   || cdvds 11794

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-id 4294  df-po 4297  df-iso 4298  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-n0 9177  df-z 9254  df-dvds 11795

This theorem is referenced by: (None)