Theorem omoe 11855

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 11832	. . . . 5 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		odd2np1 11832	. . . . 5 |- ( B e. ZZ -> ( -. 2 || B <-> E. b e. ZZ ( ( 2 x. b ) + 1 ) = B ) )
3	1, 2	bi2anan9 601	. . . 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 2639	. . . . 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 9240	. . . . . . . . 9 |- 2 e. ZZ
6		zsubcl 9253	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a - b ) e. ZZ )
7		dvdsmul1 11775	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ ( a - b ) e. ZZ ) -> 2 || ( 2 x. ( a - b ) ) )
8	5, 6, 7	sylancr 412	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( 2 x. ( a - b ) ) )
9		zcn 9217	. . . . . . . . 9 |- ( a e. ZZ -> a e. CC )
10		zcn 9217	. . . . . . . . 9 |- ( b e. ZZ -> b e. CC )
11		2cn 8949	. . . . . . . . . . . 12 |- 2 e. CC
12		mulcl 7901	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ a e. CC ) -> ( 2 x. a ) e. CC )
13	11, 12	mpan 422	. . . . . . . . . . 11 |- ( a e. CC -> ( 2 x. a ) e. CC )
14		mulcl 7901	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ b e. CC ) -> ( 2 x. b ) e. CC )
15	11, 14	mpan 422	. . . . . . . . . . 11 |- ( b e. CC -> ( 2 x. b ) e. CC )
16		ax-1cn 7867	. . . . . . . . . . . 12 |- 1 e. CC
17		pnpcan2 8159	. . . . . . . . . . . 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 1321	. . . . . . . . . . 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 287	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
20		subdi 8304	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
21	11, 20	mp3an1 1319	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
22	19, 21	eqtr4d 2206	. . . . . . . . 9 |- ( ( a e. CC /\ b e. CC ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( 2 x. ( a - b ) ) )
23	9, 10, 22	syl2an 287	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( 2 x. ( a - b ) ) )
24	8, 23	breqtrrd 4017	. . . . . . 7 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) )
25		oveq12 5862	. . . . . . . 8 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( A - B ) )
26	25	breq2d 4001	. . . . . . 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 154	. . . . . 6 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> 2 || ( A - B ) ) )
28	27	rexlimivv 2593	. . . . 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 134	. . . 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 162	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( -. 2 || A /\ -. 2 || B ) -> 2 || ( A - B ) ) )
31	30	imp 123	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( -. 2 || A /\ -. 2 || B ) ) -> 2 || ( A - B ) )
32	31	an4s 583	1 |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) -> 2 || ( A - B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   CCcc 7772   1c1 7775   + caddc 7777   x. cmul 7779   - cmin 8090   2c2 8929   ZZcz 9212   || cdvds 11749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-xor 1371  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-dvds 11750

This theorem is referenced by:  oddprm  12213  pythagtriplem13  12230