Theorem omoe 12240

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 12217	. . . . 5 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		odd2np1 12217	. . . . 5 |- ( B e. ZZ -> ( -. 2 || B <-> E. b e. ZZ ( ( 2 x. b ) + 1 ) = B ) )
3	1, 2	bi2anan9 606	. . . 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 2676	. . . . 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 9402	. . . . . . . . 9 |- 2 e. ZZ
6		zsubcl 9415	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a - b ) e. ZZ )
7		dvdsmul1 12157	. . . . . . . . 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 9379	. . . . . . . . 9 |- ( a e. ZZ -> a e. CC )
10		zcn 9379	. . . . . . . . 9 |- ( b e. ZZ -> b e. CC )
11		2cn 9109	. . . . . . . . . . . 12 |- 2 e. CC
12		mulcl 8054	. . . . . . . . . . . 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 8054	. . . . . . . . . . . 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 8020	. . . . . . . . . . . 12 |- 1 e. CC
17		pnpcan2 8314	. . . . . . . . . . . 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 1339	. . . . . . . . . . 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 8459	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
21	11, 20	mp3an1 1337	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a - b ) ) = ( ( 2 x. a ) - ( 2 x. b ) ) )
22	19, 21	eqtr4d 2241	. . . . . . . . 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 4073	. . . . . . 7 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) )
25		oveq12 5955	. . . . . . . 8 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> ( ( ( 2 x. a ) + 1 ) - ( ( 2 x. b ) + 1 ) ) = ( A - B ) )
26	25	breq2d 4057	. . . . . . 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 2629	. . . . 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 588	1 |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) -> 2 || ( A - B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   E.wrex 2485   class class class wbr 4045  (class class class)co 5946   CCcc 7925   1c1 7928   + caddc 7930   x. cmul 7932   - cmin 8245   2c2 9089   ZZcz 9374   || cdvds 12131

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-xor 1396  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-id 4341  df-po 4344  df-iso 4345  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-n0 9298  df-z 9375  df-dvds 12132

This theorem is referenced by:  oddprm  12615  pythagtriplem13  12632  gausslemma2dlem1a  15568  lgsquad2lem1  15591  lgsquad3  15594