Theorem opoe 11914

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 11892	. . . . 5 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		odd2np1 11892	. . . . 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 2657	. . . . 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 9295	. . . . . . . . 9 |- 2 e. ZZ
6		zaddcl 9307	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a + b ) e. ZZ )
7	6	peano2zd 9392	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( a + b ) + 1 ) e. ZZ )
8		dvdsmul1 11834	. . . . . . . . 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 9272	. . . . . . . . 9 |- ( a e. ZZ -> a e. CC )
11		zcn 9272	. . . . . . . . 9 |- ( b e. ZZ -> b e. CC )
12		addcl 7950	. . . . . . . . . . . . 13 |- ( ( a e. CC /\ b e. CC ) -> ( a + b ) e. CC )
13		2cn 9004	. . . . . . . . . . . . . 14 |- 2 e. CC
14		ax-1cn 7918	. . . . . . . . . . . . . 14 |- 1 e. CC
15		adddi 7957	. . . . . . . . . . . . . 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 1338	. . . . . . . . . . . . 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 7957	. . . . . . . . . . . . . 14 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a + b ) ) = ( ( 2 x. a ) + ( 2 x. b ) ) )
19	13, 18	mp3an1 1334	. . . . . . . . . . . . 13 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a + b ) ) = ( ( 2 x. a ) + ( 2 x. b ) ) )
20	19	oveq1d 5903	. . . . . . . . . . . 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 2220	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 2 x. 1 ) ) )
22		2t1e2 9086	. . . . . . . . . . . . 13 |- ( 2 x. 1 ) = 2
23		df-2 8992	. . . . . . . . . . . . 13 |- 2 = ( 1 + 1 )
24	22, 23	eqtri 2208	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = ( 1 + 1 )
25	24	oveq2i 5899	. . . . . . . . . . 11 |- ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 2 x. 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) )
26	21, 25	eqtrdi 2236	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) ) )
27		mulcl 7952	. . . . . . . . . . . 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 7952	. . . . . . . . . . . 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 8132	. . . . . . . . . . . 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 2220	. . . . . . . . 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 4041	. . . . . . 7 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) )
37		oveq12 5897	. . . . . . . 8 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) = ( A + B ) )
38	37	breq2d 4027	. . . . . . 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 2610	. . . . 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 1363   e. wcel 2158   E.wrex 2466   class class class wbr 4015  (class class class)co 5888   CCcc 7823   1c1 7826   + caddc 7828   x. cmul 7830   2c2 8984   ZZcz 9267   || cdvds 11808

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-xor 1386  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-n0 9191  df-z 9268  df-dvds 11809

This theorem is referenced by:  pythagtriplem11  12288