Theorem zneo 9145

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 9140	. . 3 |- -. ( 1 / 2 ) e. ZZ
2		2cn 8784	. . . . . . 7 |- 2 e. CC
3		zcn 9052	. . . . . . . 8 |- ( A e. ZZ -> A e. CC )
4	3	adantr 274	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> A e. CC )
5		mulcl 7740	. . . . . . 7 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
6	2, 4, 5	sylancr 410	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) e. CC )
7		zcn 9052	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
8	7	adantl 275	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> B e. CC )
9		mulcl 7740	. . . . . . 7 |- ( ( 2 e. CC /\ B e. CC ) -> ( 2 x. B ) e. CC )
10	2, 8, 9	sylancr 410	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. B ) e. CC )
11		1cnd 7775	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> 1 e. CC )
12	6, 10, 11	subaddd 8084	. . . . 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 8169	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. ( A - B ) ) = ( ( 2 x. A ) - ( 2 x. B ) ) )
15	14	oveq1d 5782	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 x. ( A - B ) ) / 2 ) = ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) )
16		zsubcl 9088	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
17		zcn 9052	. . . . . . . . . 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 8806	. . . . . . . . . 10 |- 2 # 0
20	19	a1i 9	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ ) -> 2 # 0 )
21	18, 13, 20	divcanap3d 8548	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 x. ( A - B ) ) / 2 ) = ( A - B ) )
22	15, 21	eqtr3d 2172	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) = ( A - B ) )
23	22, 16	eqeltrd 2214	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) e. ZZ )
24		oveq1 5774	. . . . . . 7 |- ( ( ( 2 x. A ) - ( 2 x. B ) ) = 1 -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) = ( 1 / 2 ) )
25	24	eleq1d 2206	. . . . . 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 154	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) = 1 -> ( 1 / 2 ) e. ZZ ) )
27	12, 26	sylbird 169	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. B ) + 1 ) = ( 2 x. A ) -> ( 1 / 2 ) e. ZZ ) )
28	27	necon3bd 2349	. . 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 2392	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) =/= ( ( 2 x. B ) + 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   =/= wne 2306   class class class wbr 3924  (class class class)co 5767   CCcc 7611   0cc0 7613   1c1 7614   + caddc 7616   x. cmul 7618   - cmin 7926   # cap 8336   / cdiv 8425   2c2 8764   ZZcz 9047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-n0 8971  df-z 9048

This theorem is referenced by:  nneo  9147  zeo2  9150