Theorem omeo 12039

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 12014	. . . . . 6 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		2z 9345	. . . . . . 7 |- 2 e. ZZ
3		divides 11932	. . . . . . 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 2664	. . . . . 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 9358	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a - b ) e. ZZ )
8		zcn 9322	. . . . . . . . . 10 |- ( a e. ZZ -> a e. CC )
9		zcn 9322	. . . . . . . . . 10 |- ( b e. ZZ -> b e. CC )
10		2cn 9053	. . . . . . . . . . . . 13 |- 2 e. CC
11		subdi 8404	. . . . . . . . . . . . 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 5933	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( ( 2 x. ( a - b ) ) + 1 ) = ( ( ( 2 x. a ) - ( 2 x. b ) ) + 1 ) )
14		mulcl 7999	. . . . . . . . . . . . 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 7999	. . . . . . . . . . . . 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 7965	. . . . . . . . . . . . 13 |- 1 e. CC
19		addsub 8230	. . . . . . . . . . . . 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 1336	. . . . . . . . . . . 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 8001	. . . . . . . . . . . . . 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 5934	. . . . . . . . . . . 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 2232	. . . . . . . . . 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 5926	. . . . . . . . . . . 12 |- ( c = ( a - b ) -> ( 2 x. c ) = ( 2 x. ( a - b ) ) )
29	28	oveq1d 5933	. . . . . . . . . . 11 |- ( c = ( a - b ) -> ( ( 2 x. c ) + 1 ) = ( ( 2 x. ( a - b ) ) + 1 ) )
30	29	eqeq1d 2202	. . . . . . . . . 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 2864	. . . . . . . . 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 5927	. . . . . . . . . 10 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( b x. 2 ) = B ) -> ( ( ( 2 x. a ) + 1 ) - ( b x. 2 ) ) = ( A - B ) )
34	33	eqeq2d 2205	. . . . . . . . 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 2495	. . . . . . . 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 2617	. . . . . 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		zsubcl 9358	. . . 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 12014	. . 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 2164   E.wrex 2473   class class class wbr 4029  (class class class)co 5918   CCcc 7870   1c1 7873   + caddc 7875   x. cmul 7877   - cmin 8190   2c2 9033   ZZcz 9317   || cdvds 11930

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-dvds 11931

This theorem is referenced by:  gausslemma2dlem1f1o  15176