Theorem omoe 10296

Description: The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
omoe	|- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) -> 2 || ( A - B ) )

Proof of Theorem omoe

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

Step	Hyp	Ref	Expression
1		odd2np1 10272	. . . . 5 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		odd2np1 10272	. . . . 5 |- ( B e. ZZ -> ( -. 2 || B <-> E. b e. ZZ ( ( 2 x. b ) + 1 ) = B ) )
3	1, 2	bi2anan9 570	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( -. 2 || A /\ -. 2 || B ) <-> ( E. a e. ZZ ( ( 2 x. a ) + 1 ) = A /\ E. b e. ZZ ( ( 2 x. b ) + 1 ) = B ) ) )
4		reeanv 2523	. . . . 5 |- ( E. a e. ZZ E. b e. ZZ ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) <-> ( E. a e. ZZ ( ( 2 x. a ) + 1 ) = A /\ E. b e. ZZ ( ( 2 x. b ) + 1 ) = B ) )
5		2z 8379	. . . . . . . . 9 |- 2 e. ZZ
6		zsubcl 8392	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a - b ) e. ZZ )
7		dvdsmul1 10217	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ ( a - b ) e. ZZ ) -> 2 || ( 2 x. ( a - b ) ) )
8	5, 6, 7	sylancr 405	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( 2 x. ( a - b ) ) )
9		zcn 8356	. . . . . . . . 9 |- ( a e. ZZ -> a e. CC )
10		zcn 8356	. . . . . . . . 9 |- ( b e. ZZ -> b e. CC )
11		2cn 8110	. . . . . . . . . . . 12 |- 2 e. CC
12		mulcl 7100	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ a e. CC ) -> ( 2 x. a ) e. CC )
13	11, 12	mpan 414	. . . . . . . . . . 11 |- ( a e. CC -> ( 2 x. a ) e. CC )
14		mulcl 7100	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ b e. CC ) -> ( 2 x. b ) e. CC )
15	11, 14	mpan 414	. . . . . . . . . . 11 |- ( b e. CC -> ( 2 x. b ) e. CC )
16		ax-1cn 7069	. . . . . . . . . . . 12 |- 1 e. CC
17		pnpcan2 7348	. . . . . . . . . . . 12 |- ( ( ( 2 x. a ) e. CC /\ ( 2 x. b ) e. CC /\ 1 e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
18	16, 17	mp3an3 1257	. . . . . . . . . . 11 |- ( ( ( 2 x. a ) e. CC /\ ( 2 x. b ) e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
19	13, 15, 18	syl2an 283	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
20		subdi 7489	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
21	11, 20	mp3an1 1255	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
22	19, 21	eqtr4d 2116	. . . . . . . . 9 |- ( ( a e. CC /\ b e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( 2 x. ( a - b ) ) )
23	9, 10, 22	syl2an 283	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( 2 x. ( a - b ) ) )
24	8, 23	breqtrrd 3811	. . . . . . 7 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) )
25		oveq12 5541	. . . . . . . 8 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( A - B ) )
26	25	breq2d 3797	. . . . . . 7 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> ( 2 || ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) <-> 2 || ( A - B ) ) )
27	24, 26	syl5ibcom 153	. . . . . 6 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> 2 || ( A - B ) ) )
28	27	rexlimivv 2482	. . . . 5 |- ( E. a e. ZZ E. b e. ZZ ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> 2 || ( A - B ) )
29	4, 28	sylbir 133	. . . 4 |- ( ( E. a e. ZZ ( ( 2 x. a ) + 1 ) = A /\ E. b e. ZZ ( ( 2 x. b ) + 1 ) = B ) -> 2 || ( A - B ) )
30	3, 29	syl6bi 161	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( -. 2 || A /\ -. 2 || B ) -> 2 || ( A - B ) ) )
31	30	imp 122	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( -. 2 || A /\ -. 2 || B ) ) -> 2 || ( A - B ) )
32	31	an4s 552	1 |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) -> 2 || ( A - B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   E.wrex 2349   class class class wbr 3785  (class class class)co 5532   CCcc 6979   1c1 6982   + caddc 6984   x. cmul 6986   - cmin 7279   2c2 8089   ZZcz 8351   || cdvds 10195

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-xor 1307  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-n0 8289  df-z 8352  df-dvds 10196

This theorem is referenced by: (None)