Theorem omeo 12539

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 12514	. . . . . 6 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		2z 9568	. . . . . . 7 |- 2 e. ZZ
3		divides 12430	. . . . . . 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 610	. . . . 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 2704	. . . . . 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 9581	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a - b ) e. ZZ )
8		zcn 9545	. . . . . . . . . 10 |- ( a e. ZZ -> a e. CC )
9		zcn 9545	. . . . . . . . . 10 |- ( b e. ZZ -> b e. CC )
10		2cn 9273	. . . . . . . . . . . . 13 |- 2 e. CC
11		subdi 8623	. . . . . . . . . . . . 13 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
12	10, 11	mp3an1 1361	. . . . . . . . . . . 12 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
13	12	oveq1d 6043	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( ( 2 x. ( a - b ) ) + 1 ) = ( ( ( 2 x. a ) - ( 2 x. b ) ) + 1 ) )
14		mulcl 8219	. . . . . . . . . . . . 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 8219	. . . . . . . . . . . . 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 8185	. . . . . . . . . . . . 13 |- 1 e. CC
19		addsub 8449	. . . . . . . . . . . . 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 1362	. . . . . . . . . . . 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 8221	. . . . . . . . . . . . . 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 6044	. . . . . . . . . . . 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 2270	. . . . . . . . . 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 6036	. . . . . . . . . . . 12 |- ( c = ( a - b ) -> ( 2 x. c ) = ( 2 x. ( a - b ) ) )
29	28	oveq1d 6043	. . . . . . . . . . 11 |- ( c = ( a - b ) -> ( ( 2 x. c ) + 1 ) = ( ( 2 x. ( a - b ) ) + 1 ) )
30	29	eqeq1d 2240	. . . . . . . . . 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 2911	. . . . . . . . 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 6037	. . . . . . . . . 10 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( b x. 2 ) = B ) -> ( ( ( 2 x. a ) + 1 ) - ( b x. 2 ) ) = ( A - B ) )
34	33	eqeq2d 2243	. . . . . . . . 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 2534	. . . . . . . 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 2657	. . . . . 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 592	. 2 |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ 2 || B ) ) -> E. c e. ZZ ( ( 2 x. c ) + 1 ) = ( A - B ) )
42		zsubcl 9581	. . . 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 12514	. . 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 1398   e. wcel 2202   E.wrex 2512   class class class wbr 4093  (class class class)co 6028   CCcc 8090   1c1 8093   + caddc 8095   x. cmul 8097   - cmin 8409   2c2 9253   ZZcz 9540   || cdvds 12428

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-n0 9462  df-z 9541  df-dvds 12429

This theorem is referenced by:  gausslemma2dlem1f1o  15879