Theorem zneo 8448

Description: No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 18-May-2014.)

Assertion

Ref	Expression
zneo	|- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) =/= ( ( 2 x. B ) + 1 ) )

Proof of Theorem zneo

Step	Hyp	Ref	Expression
1		halfnz 8443	. . 3 |- -. ( 1 / 2 ) e. ZZ
2		2cn 8110	. . . . . . 7 |- 2 e. CC
3		zcn 8356	. . . . . . . 8 |- ( A e. ZZ -> A e. CC )
4	3	adantr 270	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> A e. CC )
5		mulcl 7100	. . . . . . 7 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
6	2, 4, 5	sylancr 405	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) e. CC )
7		zcn 8356	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
8	7	adantl 271	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> B e. CC )
9		mulcl 7100	. . . . . . 7 |- ( ( 2 e. CC /\ B e. CC ) -> ( 2 x. B ) e. CC )
10	2, 8, 9	sylancr 405	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. B ) e. CC )
11		1cnd 7135	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> 1 e. CC )
12	6, 10, 11	subaddd 7437	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) = 1 <-> ( ( 2 x. B ) + 1 ) = ( 2 x. A ) ) )
13	2	a1i 9	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> 2 e. CC )
14	13, 4, 8	subdid 7518	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. ( A - B ) ) = ( ( 2 x. A ) - ( 2 x. B ) ) )
15	14	oveq1d 5547	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 x. ( A - B ) ) / 2 ) = ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) )
16		zsubcl 8392	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
17		zcn 8356	. . . . . . . . . 10 |- ( ( A - B ) e. ZZ -> ( A - B ) e. CC )
18	16, 17	syl 14	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. CC )
19		2ap0 8132	. . . . . . . . . 10 |- 2 # 0
20	19	a1i 9	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ ) -> 2 # 0 )
21	18, 13, 20	divcanap3d 7882	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 x. ( A - B ) ) / 2 ) = ( A - B ) )
22	15, 21	eqtr3d 2115	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) = ( A - B ) )
23	22, 16	eqeltrd 2155	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) e. ZZ )
24		oveq1 5539	. . . . . . 7 |- ( ( ( 2 x. A ) - ( 2 x. B ) ) = 1 -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) = ( 1 / 2 ) )
25	24	eleq1d 2147	. . . . . 6 |- ( ( ( 2 x. A ) - ( 2 x. B ) ) = 1 -> ( ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) e. ZZ <-> ( 1 / 2 ) e. ZZ ) )
26	23, 25	syl5ibcom 153	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) = 1 -> ( 1 / 2 ) e. ZZ ) )
27	12, 26	sylbird 168	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. B ) + 1 ) = ( 2 x. A ) -> ( 1 / 2 ) e. ZZ ) )
28	27	necon3bd 2288	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( -. ( 1 / 2 ) e. ZZ -> ( ( 2 x. B ) + 1 ) =/= ( 2 x. A ) ) )
29	1, 28	mpi 15	. 2 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 x. B ) + 1 ) =/= ( 2 x. A ) )
30	29	necomd 2331	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) =/= ( ( 2 x. B ) + 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   =/= wne 2245   class class class wbr 3785  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   - cmin 7279   # cap 7681   / cdiv 7760   2c2 8089   ZZcz 8351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-n0 8289  df-z 8352

This theorem is referenced by:  nneo  8450  zeo2  8453