Theorem zneo 9292

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 9287	. . 3 |- -. ( 1 / 2 ) e. ZZ
2		2cn 8928	. . . . . . 7 |- 2 e. CC
3		zcn 9196	. . . . . . . 8 |- ( A e. ZZ -> A e. CC )
4	3	adantr 274	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> A e. CC )
5		mulcl 7880	. . . . . . 7 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
6	2, 4, 5	sylancr 411	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) e. CC )
7		zcn 9196	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
8	7	adantl 275	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> B e. CC )
9		mulcl 7880	. . . . . . 7 |- ( ( 2 e. CC /\ B e. CC ) -> ( 2 x. B ) e. CC )
10	2, 8, 9	sylancr 411	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. B ) e. CC )
11		1cnd 7915	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> 1 e. CC )
12	6, 10, 11	subaddd 8227	. . . . 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 8312	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. ( A - B ) ) = ( ( 2 x. A ) - ( 2 x. B ) ) )
15	14	oveq1d 5857	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 x. ( A - B ) ) / 2 ) = ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) )
16		zsubcl 9232	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
17		zcn 9196	. . . . . . . . . 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 8950	. . . . . . . . . 10 |- 2 # 0
20	19	a1i 9	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ ) -> 2 # 0 )
21	18, 13, 20	divcanap3d 8691	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 x. ( A - B ) ) / 2 ) = ( A - B ) )
22	15, 21	eqtr3d 2200	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) = ( A - B ) )
23	22, 16	eqeltrd 2243	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) e. ZZ )
24		oveq1 5849	. . . . . . 7 |- ( ( ( 2 x. A ) - ( 2 x. B ) ) = 1 -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) = ( 1 / 2 ) )
25	24	eleq1d 2235	. . . . . 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 2379	. . 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 2422	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) =/= ( ( 2 x. B ) + 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   - cmin 8069   # cap 8479   / cdiv 8568   2c2 8908   ZZcz 9191

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192

This theorem is referenced by:  nneo  9294  zeo2  9297