Theorem omoe 12582

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 12559	. . . . 5 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		odd2np1 12559	. . . . 5 |- ( B e. ZZ -> ( -. 2 || B <-> E. b e. ZZ ( ( 2 x. b ) + 1 ) = B ) )
3	1, 2	bi2anan9 610	. . . 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 2713	. . . . 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 9605	. . . . . . . . 9 |- 2 e. ZZ
6		zsubcl 9618	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a - b ) e. ZZ )
7		dvdsmul1 12499	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ ( a - b ) e. ZZ ) -> 2 || ( 2 x. ( a - b ) ) )
8	5, 6, 7	sylancr 414	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( 2 x. ( a - b ) ) )
9		zcn 9582	. . . . . . . . 9 |- ( a e. ZZ -> a e. CC )
10		zcn 9582	. . . . . . . . 9 |- ( b e. ZZ -> b e. CC )
11		2cn 9308	. . . . . . . . . . . 12 |- 2 e. CC
12		mulcl 8254	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ a e. CC ) -> ( 2 x. a ) e. CC )
13	11, 12	mpan 424	. . . . . . . . . . 11 |- ( a e. CC -> ( 2 x. a ) e. CC )
14		mulcl 8254	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ b e. CC ) -> ( 2 x. b ) e. CC )
15	11, 14	mpan 424	. . . . . . . . . . 11 |- ( b e. CC -> ( 2 x. b ) e. CC )
16		ax-1cn 8220	. . . . . . . . . . . 12 |- 1 e. CC
17		pnpcan2 8513	. . . . . . . . . . . 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 1363	. . . . . . . . . . 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 289	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
20		subdi 8658	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
21	11, 20	mp3an1 1361	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
22	19, 21	eqtr4d 2268	. . . . . . . . 9 |- ( ( a e. CC /\ b e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( 2 x. ( a - b ) ) )
23	9, 10, 22	syl2an 289	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( 2 x. ( a - b ) ) )
24	8, 23	breqtrrd 4137	. . . . . . 7 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) )
25		oveq12 6059	. . . . . . . 8 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( A - B ) )
26	25	breq2d 4121	. . . . . . 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 155	. . . . . 6 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> 2 || ( A - B ) ) )
28	27	rexlimivv 2666	. . . . 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 135	. . . 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	biimtrdi 163	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( -. 2 || A /\ -. 2 || B ) -> 2 || ( A - B ) ) )
31	30	imp 124	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( -. 2 || A /\ -. 2 || B ) ) -> 2 || ( A - B ) )
32	31	an4s 592	1 |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) -> 2 || ( A - B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4109  (class class class)co 6050   CCcc 8125   1c1 8128   + caddc 8130   x. cmul 8132   - cmin 8444   2c2 9288   ZZcz 9577   || cdvds 12473

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-dvds 12474

This theorem is referenced by:  oddprm  12957  pythagtriplem13  12974  gausslemma2dlem1a  15931  lgsquad2lem1  15954  lgsquad3  15957