Theorem opoe 12039

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 12017	. . . . 5 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		odd2np1 12017	. . . . 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 2664	. . . . 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 9348	. . . . . . . . 9 |- 2 e. ZZ
6		zaddcl 9360	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a + b ) e. ZZ )
7	6	peano2zd 9445	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( a + b ) + 1 ) e. ZZ )
8		dvdsmul1 11959	. . . . . . . . 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 9325	. . . . . . . . 9 |- ( a e. ZZ -> a e. CC )
11		zcn 9325	. . . . . . . . 9 |- ( b e. ZZ -> b e. CC )
12		addcl 7999	. . . . . . . . . . . . 13 |- ( ( a e. CC /\ b e. CC ) -> ( a + b ) e. CC )
13		2cn 9055	. . . . . . . . . . . . . 14 |- 2 e. CC
14		ax-1cn 7967	. . . . . . . . . . . . . 14 |- 1 e. CC
15		adddi 8006	. . . . . . . . . . . . . 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 8006	. . . . . . . . . . . . . 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 5934	. . . . . . . . . . . 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 2226	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 2 x. 1 ) ) )
22		2t1e2 9138	. . . . . . . . . . . . 13 |- ( 2 x. 1 ) = 2
23		df-2 9043	. . . . . . . . . . . . 13 |- 2 = ( 1 + 1 )
24	22, 23	eqtri 2214	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = ( 1 + 1 )
25	24	oveq2i 5930	. . . . . . . . . . 11 |- ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 2 x. 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) )
26	21, 25	eqtrdi 2242	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) ) )
27		mulcl 8001	. . . . . . . . . . . 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 8001	. . . . . . . . . . . 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 8182	. . . . . . . . . . . 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 2226	. . . . . . . . 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 4056	. . . . . . 7 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) )
37		oveq12 5928	. . . . . . . 8 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) = ( A + B ) )
38	37	breq2d 4042	. . . . . . 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 2617	. . . . 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 2164   E.wrex 2473   class class class wbr 4030  (class class class)co 5919   CCcc 7872   1c1 7875   + caddc 7877   x. cmul 7879   2c2 9035   ZZcz 9320   || cdvds 11933

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-dvds 11934

This theorem is referenced by:  pythagtriplem11  12415