Theorem opoe 12060

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

Assertion

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

Proof of Theorem opoe

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

Step	Hyp	Ref	Expression
1		odd2np1 12038	. . . . 5 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		odd2np1 12038	. . . . 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 2667	. . . . 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 9354	. . . . . . . . 9 |- 2 e. ZZ
6		zaddcl 9366	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a + b ) e. ZZ )
7	6	peano2zd 9451	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( a + b ) + 1 ) e. ZZ )
8		dvdsmul1 11978	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ ( ( a + b ) + 1 ) e. ZZ ) -> 2 || ( 2 x. ( ( a + b ) + 1 ) ) )
9	5, 7, 8	sylancr 414	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( 2 x. ( ( a + b ) + 1 ) ) )
10		zcn 9331	. . . . . . . . 9 |- ( a e. ZZ -> a e. CC )
11		zcn 9331	. . . . . . . . 9 |- ( b e. ZZ -> b e. CC )
12		addcl 8004	. . . . . . . . . . . . 13 |- ( ( a e. CC /\ b e. CC ) -> ( a + b ) e. CC )
13		2cn 9061	. . . . . . . . . . . . . 14 |- 2 e. CC
14		ax-1cn 7972	. . . . . . . . . . . . . 14 |- 1 e. CC
15		adddi 8011	. . . . . . . . . . . . . 14 |- ( ( 2 e. CC /\ ( a + b ) e. CC /\ 1 e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( 2 x. ( a + b ) ) + ( 2 x. 1 ) ) )
16	13, 14, 15	mp3an13 1339	. . . . . . . . . . . . 13 |- ( ( a + b ) e. CC -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( 2 x. ( a + b ) ) + ( 2 x. 1 ) ) )
17	12, 16	syl 14	. . . . . . . . . . . 12 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( 2 x. ( a + b ) ) + ( 2 x. 1 ) ) )
18		adddi 8011	. . . . . . . . . . . . . 14 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a + b ) ) = ( ( 2 x. a ) + ( 2 x. b ) ) )
19	13, 18	mp3an1 1335	. . . . . . . . . . . . 13 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a + b ) ) = ( ( 2 x. a ) + ( 2 x. b ) ) )
20	19	oveq1d 5937	. . . . . . . . . . . 12 |- ( ( a e. CC /\ b e. CC ) -> ( ( 2 x. ( a + b ) ) + ( 2 x. 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 2 x. 1 ) ) )
21	17, 20	eqtrd 2229	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 2 x. 1 ) ) )
22		2t1e2 9144	. . . . . . . . . . . . 13 |- ( 2 x. 1 ) = 2
23		df-2 9049	. . . . . . . . . . . . 13 |- 2 = ( 1 + 1 )
24	22, 23	eqtri 2217	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = ( 1 + 1 )
25	24	oveq2i 5933	. . . . . . . . . . 11 |- ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 2 x. 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) )
26	21, 25	eqtrdi 2245	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) ) )
27		mulcl 8006	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ a e. CC ) -> ( 2 x. a ) e. CC )
28	13, 27	mpan 424	. . . . . . . . . . 11 |- ( a e. CC -> ( 2 x. a ) e. CC )
29		mulcl 8006	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ b e. CC ) -> ( 2 x. b ) e. CC )
30	13, 29	mpan 424	. . . . . . . . . . 11 |- ( b e. CC -> ( 2 x. b ) e. CC )
31		add4 8187	. . . . . . . . . . . 12 |- ( ( ( ( 2 x. a ) e. CC /\ ( 2 x. b ) e. CC ) /\ ( 1 e. CC /\ 1 e. CC ) ) -> ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) ) = ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) )
32	14, 14, 31	mpanr12 439	. . . . . . . . . . 11 |- ( ( ( 2 x. a ) e. CC /\ ( 2 x. b ) e. CC ) -> ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) ) = ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) )
33	28, 30, 32	syl2an 289	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) ) = ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) )
34	26, 33	eqtrd 2229	. . . . . . . . 9 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) )
35	10, 11, 34	syl2an 289	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) )
36	9, 35	breqtrd 4059	. . . . . . 7 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) )
37		oveq12 5931	. . . . . . . 8 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) = ( A + B ) )
38	37	breq2d 4045	. . . . . . 7 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> ( 2 || ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) <-> 2 || ( A + B ) ) )
39	36, 38	syl5ibcom 155	. . . . . 6 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> 2 || ( A + B ) ) )
40	39	rexlimivv 2620	. . . . 5 |- ( E. a e. ZZ E. b e. ZZ ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> 2 || ( A + B ) )
41	4, 40	sylbir 135	. . . 4 |- ( ( E. a e. ZZ ( ( 2 x. a ) + 1 ) = A /\ E. b e. ZZ ( ( 2 x. b ) + 1 ) = B ) -> 2 || ( A + B ) )
42	3, 41	biimtrdi 163	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( -. 2 || A /\ -. 2 || B ) -> 2 || ( A + B ) ) )
43	42	imp 124	. 2 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ ( -. 2 || A /\ -. 2 || B ) ) -> 2 || ( A + B ) )
44	43	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 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4033  (class class class)co 5922   CCcc 7877   1c1 7880   + caddc 7882   x. cmul 7884   2c2 9041   ZZcz 9326   || cdvds 11952

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-xor 1387  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-dvds 11953

This theorem is referenced by:  pythagtriplem11  12443