Theorem opeo 12062

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 12038	. . . . . 6 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		2z 9354	. . . . . . 7 |- 2 e. ZZ
3		divides 11954	. . . . . . 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 2667	. . . . . 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 9366	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a + b ) e. ZZ )
8		zcn 9331	. . . . . . . . . 10 |- ( a e. ZZ -> a e. CC )
9		zcn 9331	. . . . . . . . . 10 |- ( b e. ZZ -> b e. CC )
10		2cn 9061	. . . . . . . . . . . . 13 |- 2 e. CC
11		adddi 8011	. . . . . . . . . . . . 13 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a + b ) ) = ( ( 2 x. a ) + ( 2 x. b ) ) )
12	10, 11	mp3an1 1335	. . . . . . . . . . . 12 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a + b ) ) = ( ( 2 x. a ) + ( 2 x. b ) ) )
13	12	oveq1d 5937	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( ( 2 x. ( a + b ) ) + 1 ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + 1 ) )
14		mulcl 8006	. . . . . . . . . . . . 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 8006	. . . . . . . . . . . . 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 7972	. . . . . . . . . . . . 13 |- 1 e. CC
19		add32 8185	. . . . . . . . . . . . 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 1337	. . . . . . . . . . . 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 8008	. . . . . . . . . . . . . 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 5938	. . . . . . . . . . 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 2233	. . . . . . . . . 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 5930	. . . . . . . . . . . 12 |- ( c = ( a + b ) -> ( 2 x. c ) = ( 2 x. ( a + b ) ) )
29	28	oveq1d 5937	. . . . . . . . . . 11 |- ( c = ( a + b ) -> ( ( 2 x. c ) + 1 ) = ( ( 2 x. ( a + b ) ) + 1 ) )
30	29	eqeq1d 2205	. . . . . . . . . 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 2868	. . . . . . . . 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 5931	. . . . . . . . . 10 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( b x. 2 ) = B ) -> ( ( ( 2 x. a ) + 1 ) + ( b x. 2 ) ) = ( A + B ) )
34	33	eqeq2d 2208	. . . . . . . . 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 2498	. . . . . . . 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 2620	. . . . . 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	biimtrdi 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 9366	. . . 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 12038	. . 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 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4033  (class class class)co 5922   CCcc 7877   1c1 7880   + caddc 7882   x. cmul 7884   2c2 9041   ZZcz 9326   || cdvds 11952

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-dvds 11953

This theorem is referenced by: (None)