Theorem opoe 12640

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 12618	. . . . 5 |- ( A e. ZZ -> ( -. 2 || A <-> E. a e. ZZ ( ( 2 x. a ) + 1 ) = A ) )
2		odd2np1 12618	. . . . 5 |- ( B e. ZZ -> ( -. 2 || B <-> E. b e. ZZ ( ( 2 x. b ) + 1 ) = B ) )
3	1, 2	bi2anan9 614	. . . 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 2721	. . . . 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 9651	. . . . . . . . 9 |- 2 e. ZZ
6		zaddcl 9663	. . . . . . . . . 10 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( a + b ) e. ZZ )
7	6	peano2zd 9750	. . . . . . . . 9 |- ( ( a e. ZZ /\ b e. ZZ ) -> ( ( a + b ) + 1 ) e. ZZ )
8		dvdsmul1 12558	. . . . . . . . 9 |- ( ( 2 e. ZZ /\ ( ( a + b ) + 1 ) e. ZZ ) -> 2 || ( 2 x. ( ( a + b ) + 1 ) ) )
9	5, 7, 8	sylancr 418	. . . . . . . 8 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( 2 x. ( ( a + b ) + 1 ) ) )
10		zcn 9628	. . . . . . . . 9 |- ( a e. ZZ -> a e. CC )
11		zcn 9628	. . . . . . . . 9 |- ( b e. ZZ -> b e. CC )
12		addcl 8294	. . . . . . . . . . . . 13 |- ( ( a e. CC /\ b e. CC ) -> ( a + b ) e. CC )
13		2cn 9354	. . . . . . . . . . . . . 14 |- 2 e. CC
14		ax-1cn 8262	. . . . . . . . . . . . . 14 |- 1 e. CC
15		adddi 8301	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 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 8301	. . . . . . . . . . . . . 14 |- ( ( 2 e. CC /\ a e. CC /\ b e. CC ) -> ( 2 x. ( a + b ) ) = ( ( 2 x. a ) + ( 2 x. b ) ) )
19	13, 18	mp3an1 1365	. . . . . . . . . . . . 13 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( a + b ) ) = ( ( 2 x. a ) + ( 2 x. b ) ) )
20	19	oveq1d 6090	. . . . . . . . . . . 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 2271	. . . . . . . . . . 11 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 2 x. 1 ) ) )
22		2t1e2 9437	. . . . . . . . . . . . 13 |- ( 2 x. 1 ) = 2
23		df-2 9342	. . . . . . . . . . . . 13 |- 2 = ( 1 + 1 )
24	22, 23	eqtri 2259	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = ( 1 + 1 )
25	24	oveq2i 6086	. . . . . . . . . . 11 |- ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 2 x. 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) )
26	21, 25	eqtrdi 2287	. . . . . . . . . 10 |- ( ( a e. CC /\ b e. CC ) -> ( 2 x. ( ( a + b ) + 1 ) ) = ( ( ( 2 x. a ) + ( 2 x. b ) ) + ( 1 + 1 ) ) )
27		mulcl 8296	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ a e. CC ) -> ( 2 x. a ) e. CC )
28	13, 27	mpan 428	. . . . . . . . . . 11 |- ( a e. CC -> ( 2 x. a ) e. CC )
29		mulcl 8296	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ b e. CC ) -> ( 2 x. b ) e. CC )
30	13, 29	mpan 428	. . . . . . . . . . 11 |- ( b e. CC -> ( 2 x. b ) e. CC )
31		add4 8477	. . . . . . . . . . . 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 443	. . . . . . . . . . 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 2271	. . . . . . . . 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 4151	. . . . . . 7 |- ( ( a e. ZZ /\ b e. ZZ ) -> 2 || ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) )
37		oveq12 6084	. . . . . . . 8 |- ( ( ( ( 2 x. a ) + 1 ) = A /\ ( ( 2 x. b ) + 1 ) = B ) -> ( ( ( 2 x. a ) + 1 ) + ( ( 2 x. b ) + 1 ) ) = ( A + B ) )
38	37	breq2d 4137	. . . . . . 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 2674	. . . . 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 596	1 |- ( ( ( A e. ZZ /\ -. 2 || A ) /\ ( B e. ZZ /\ -. 2 || B ) ) -> 2 || ( A + B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   CCcc 8167   1c1 8170   + caddc 8172   x. cmul 8174   2c2 9334   ZZcz 9623   || cdvds 12532

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-dvds 12533

This theorem is referenced by:  pythagtriplem11  13031  eupth2lem3lem4fi  16628