Theorem omeo 12409

Description: The difference 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
omeo	|- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) -> -. 2 || ( A - B ) )

Proof of Theorem omeo

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		odd2np1 12384	. . . . . 6 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		2z 9474	. . . . . . 7 |- 2 e. ZZ
3		divides 12300	. . . . . . 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 608	. . . . 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 2701	. . . . . 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		zsubcl 9487	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a - b ) e. ZZ )
8		zcn 9451	. . . . . . . . . 10 |- ( a e. ZZ -> a e. CC )
9		zcn 9451	. . . . . . . . . 10 |- ( b e. ZZ -> b e. CC )
10		2cn 9181	. . . . . . . . . . . . 13 |- 2 e. CC
11		subdi 8531	. . . . . . . . . . . . 13 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
12	10, 11	mp3an1 1358	. . . . . . . . . . . 12 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
13	12	oveq1d 6016	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( ( 2 x. ( a - b ) ) + 1 ) = ( ( ( 2 x. a ) - ( 2 x. b ) ) + 1 ) )
14		mulcl 8126	. . . . . . . . . . . . 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 8126	. . . . . . . . . . . . 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 8092	. . . . . . . . . . . . 13 |- 1 e. CC
19		addsub 8357	. . . . . . . . . . . . 13 |- ( ( ( 2 x. a ) e. CC /\ 1 e. CC /\ ( 2 x. b ) e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( 2 x. b ) ) = ( ( ( 2 x. a ) - ( 2 x. b ) ) + 1 ) )
20	18, 19	mp3an2 1359	. . . . . . . . . . . 12 |- ( ( ( 2 x. a ) e. CC /\ ( 2 x. b ) e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( 2 x. b ) ) = ( ( ( 2 x. a ) - ( 2 x. b ) ) + 1 ) )
21	15, 17, 20	syl2an 289	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( 2 x. b ) ) = ( ( ( 2 x. a ) - ( 2 x. b ) ) + 1 ) )
22		mulcom 8128	. . . . . . . . . . . . . 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	oveq2d 6017	. . . . . . . . . . . 12 |- ( b e. CC -> ( ( ( 2 x. a ) + 1 ) - ( 2 x. b ) ) = ( ( ( 2 x. a ) + 1 ) - ( b x. 2 ) ) )
25	24	adantl 277	. . . . . . . . . . 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	3eqtr2d 2268	. . . . . . . . . 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 6009	. . . . . . . . . . . 12 |- ( c = ( a - b ) -> ( 2 x. c ) = ( 2 x. ( a - b ) ) )
29	28	oveq1d 6016	. . . . . . . . . . 11 |- ( c = ( a - b ) -> ( ( 2 x. c ) + 1 ) = ( ( 2 x. ( a - b ) ) + 1 ) )
30	29	eqeq1d 2238	. . . . . . . . . 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 2907	. . . . . . . . 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 6010	. . . . . . . . . 10 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( b x. 2 ) = B ) -> ( ( ( 2 x. a ) + 1 ) - ( b x. 2 ) ) = ( A - B ) )
34	33	eqeq2d 2241	. . . . . . . . 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 2531	. . . . . . . 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 2654	. . . . . 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 590	. 2 |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) -> E. c e. ZZ ( ( 2 x. c ) + 1 ) = ( A - B ) )
42		zsubcl 9487	. . . 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 12384	. . 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 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4083  (class class class)co 6001   CCcc 7997   1c1 8000   + caddc 8002   x. cmul 8004   - cmin 8317   2c2 9161   ZZcz 9446   || cdvds 12298

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-dvds 12299

This theorem is referenced by:  gausslemma2dlem1f1o  15739