Theorem zneo 9548

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 9543	. . 3 |- -. ( 1 / 2 ) e. ZZ
2		2cn 9181	. . . . . . 7 |- 2 e. CC
3		zcn 9451	. . . . . . . 8 |- ( A e. ZZ -> A e. CC )
4	3	adantr 276	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> A e. CC )
5		mulcl 8126	. . . . . . 7 |- ( ( 2 e. CC /\ A e. CC ) -> ( 2 x. A ) e. CC )
6	2, 4, 5	sylancr 414	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) e. CC )
7		zcn 9451	. . . . . . . 8 |- ( B e. ZZ -> B e. CC )
8	7	adantl 277	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> B e. CC )
9		mulcl 8126	. . . . . . 7 |- ( ( 2 e. CC /\ B e. CC ) -> ( 2 x. B ) e. CC )
10	2, 8, 9	sylancr 414	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. B ) e. CC )
11		1cnd 8162	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> 1 e. CC )
12	6, 10, 11	subaddd 8475	. . . . 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 8560	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. ( A - B ) ) = ( ( 2 x. A ) - ( 2 x. B ) ) )
15	14	oveq1d 6016	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 x. ( A - B ) ) / 2 ) = ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) )
16		zsubcl 9487	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
17		zcn 9451	. . . . . . . . . 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 9203	. . . . . . . . . 10 |- 2 # 0
20	19	a1i 9	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ ) -> 2 # 0 )
21	18, 13, 20	divcanap3d 8942	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 x. ( A - B ) ) / 2 ) = ( A - B ) )
22	15, 21	eqtr3d 2264	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) = ( A - B ) )
23	22, 16	eqeltrd 2306	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) e. ZZ )
24		oveq1 6008	. . . . . . 7 |- ( ( ( 2 x. A ) - ( 2 x. B ) ) = 1 -> ( ( ( 2 x. A ) - ( 2 x. B ) ) / 2 ) = ( 1 / 2 ) )
25	24	eleq1d 2298	. . . . . 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 155	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. A ) - ( 2 x. B ) ) = 1 -> ( 1 / 2 ) e. ZZ ) )
27	12, 26	sylbird 170	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( 2 x. B ) + 1 ) = ( 2 x. A ) -> ( 1 / 2 ) e. ZZ ) )
28	27	necon3bd 2443	. . 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 2486	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 2 x. A ) =/= ( ( 2 x. B ) + 1 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4083  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   - cmin 8317   # cap 8728   / cdiv 8819   2c2 9161   ZZcz 9446

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447

This theorem is referenced by:  nneo  9550  zeo2  9553