Theorem nneoor 9581

Description: A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.)

Assertion

Ref	Expression
nneoor	|- ( N e. NN -> ( ( N / 2 ) e. NN \/ ( ( N + 1 ) / 2 ) e. NN ) )

Proof of Theorem nneoor

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6024	. . . . . 6 |- ( j = 1 -> ( j + 1 ) = ( 1 + 1 ) )
2	1	oveq1d 6032	. . . . 5 |- ( j = 1 -> ( ( j + 1 ) / 2 ) = ( ( 1 + 1 ) / 2 ) )
3	2	eleq1d 2300	. . . 4 |- ( j = 1 -> ( ( ( j + 1 ) / 2 ) e. NN <-> ( ( 1 + 1 ) / 2 ) e. NN ) )
4		oveq1 6024	. . . . 5 |- ( j = 1 -> ( j / 2 ) = ( 1 / 2 ) )
5	4	eleq1d 2300	. . . 4 |- ( j = 1 -> ( ( j / 2 ) e. NN <-> ( 1 / 2 ) e. NN ) )
6	3, 5	orbi12d 800	. . 3 |- ( j = 1 -> ( ( ( ( j + 1 ) / 2 ) e. NN \/ ( j / 2 ) e. NN ) <-> ( ( ( 1 + 1 ) / 2 ) e. NN \/ ( 1 / 2 ) e. NN ) ) )
7		oveq1 6024	. . . . . 6 |- ( j = k -> ( j + 1 ) = ( k + 1 ) )
8	7	oveq1d 6032	. . . . 5 |- ( j = k -> ( ( j + 1 ) / 2 ) = ( ( k + 1 ) / 2 ) )
9	8	eleq1d 2300	. . . 4 |- ( j = k -> ( ( ( j + 1 ) / 2 ) e. NN <-> ( ( k + 1 ) / 2 ) e. NN ) )
10		oveq1 6024	. . . . 5 |- ( j = k -> ( j / 2 ) = ( k / 2 ) )
11	10	eleq1d 2300	. . . 4 |- ( j = k -> ( ( j / 2 ) e. NN <-> ( k / 2 ) e. NN ) )
12	9, 11	orbi12d 800	. . 3 |- ( j = k -> ( ( ( ( j + 1 ) / 2 ) e. NN \/ ( j / 2 ) e. NN ) <-> ( ( ( k + 1 ) / 2 ) e. NN \/ ( k / 2 ) e. NN ) ) )
13		oveq1 6024	. . . . . 6 |- ( j = ( k + 1 ) -> ( j + 1 ) = ( ( k + 1 ) + 1 ) )
14	13	oveq1d 6032	. . . . 5 |- ( j = ( k + 1 ) -> ( ( j + 1 ) / 2 ) = ( ( ( k + 1 ) + 1 ) / 2 ) )
15	14	eleq1d 2300	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( j + 1 ) / 2 ) e. NN <-> ( ( ( k + 1 ) + 1 ) / 2 ) e. NN ) )
16		oveq1 6024	. . . . 5 |- ( j = ( k + 1 ) -> ( j / 2 ) = ( ( k + 1 ) / 2 ) )
17	16	eleq1d 2300	. . . 4 |- ( j = ( k + 1 ) -> ( ( j / 2 ) e. NN <-> ( ( k + 1 ) / 2 ) e. NN ) )
18	15, 17	orbi12d 800	. . 3 |- ( j = ( k + 1 ) -> ( ( ( ( j + 1 ) / 2 ) e. NN \/ ( j / 2 ) e. NN ) <-> ( ( ( ( k + 1 ) + 1 ) / 2 ) e. NN \/ ( ( k + 1 ) / 2 ) e. NN ) ) )
19		oveq1 6024	. . . . . 6 |- ( j = N -> ( j + 1 ) = ( N + 1 ) )
20	19	oveq1d 6032	. . . . 5 |- ( j = N -> ( ( j + 1 ) / 2 ) = ( ( N + 1 ) / 2 ) )
21	20	eleq1d 2300	. . . 4 |- ( j = N -> ( ( ( j + 1 ) / 2 ) e. NN <-> ( ( N + 1 ) / 2 ) e. NN ) )
22		oveq1 6024	. . . . 5 |- ( j = N -> ( j / 2 ) = ( N / 2 ) )
23	22	eleq1d 2300	. . . 4 |- ( j = N -> ( ( j / 2 ) e. NN <-> ( N / 2 ) e. NN ) )
24	21, 23	orbi12d 800	. . 3 |- ( j = N -> ( ( ( ( j + 1 ) / 2 ) e. NN \/ ( j / 2 ) e. NN ) <-> ( ( ( N + 1 ) / 2 ) e. NN \/ ( N / 2 ) e. NN ) ) )
25		df-2 9201	. . . . . . 7 |- 2 = ( 1 + 1 )
26	25	oveq1i 6027	. . . . . 6 |- ( 2 / 2 ) = ( ( 1 + 1 ) / 2 )
27		2div2e1 9275	. . . . . 6 |- ( 2 / 2 ) = 1
28	26, 27	eqtr3i 2254	. . . . 5 |- ( ( 1 + 1 ) / 2 ) = 1
29		1nn 9153	. . . . 5 |- 1 e. NN
30	28, 29	eqeltri 2304	. . . 4 |- ( ( 1 + 1 ) / 2 ) e. NN
31	30	orci 738	. . 3 |- ( ( ( 1 + 1 ) / 2 ) e. NN \/ ( 1 / 2 ) e. NN )
32		peano2nn 9154	. . . . . 6 |- ( ( k / 2 ) e. NN -> ( ( k / 2 ) + 1 ) e. NN )
33		nncn 9150	. . . . . . . 8 |- ( k e. NN -> k e. CC )
34		add1p1 9393	. . . . . . . . . 10 |- ( k e. CC -> ( ( k + 1 ) + 1 ) = ( k + 2 ) )
35	34	oveq1d 6032	. . . . . . . . 9 |- ( k e. CC -> ( ( ( k + 1 ) + 1 ) / 2 ) = ( ( k + 2 ) / 2 ) )
36		2cn 9213	. . . . . . . . . . 11 |- 2 e. CC
37		2ap0 9235	. . . . . . . . . . . 12 |- 2 # 0
38	36, 37	pm3.2i 272	. . . . . . . . . . 11 |- ( 2 e. CC /\ 2 # 0 )
39		divdirap 8876	. . . . . . . . . . 11 |- ( ( k e. CC /\ 2 e. CC /\ ( 2 e. CC /\ 2 # 0 ) ) -> ( ( k + 2 ) / 2 ) = ( ( k / 2 ) + ( 2 / 2 ) ) )
40	36, 38, 39	mp3an23 1365	. . . . . . . . . 10 |- ( k e. CC -> ( ( k + 2 ) / 2 ) = ( ( k / 2 ) + ( 2 / 2 ) ) )
41	27	oveq2i 6028	. . . . . . . . . 10 |- ( ( k / 2 ) + ( 2 / 2 ) ) = ( ( k / 2 ) + 1 )
42	40, 41	eqtrdi 2280	. . . . . . . . 9 |- ( k e. CC -> ( ( k + 2 ) / 2 ) = ( ( k / 2 ) + 1 ) )
43	35, 42	eqtrd 2264	. . . . . . . 8 |- ( k e. CC -> ( ( ( k + 1 ) + 1 ) / 2 ) = ( ( k / 2 ) + 1 ) )
44	33, 43	syl 14	. . . . . . 7 |- ( k e. NN -> ( ( ( k + 1 ) + 1 ) / 2 ) = ( ( k / 2 ) + 1 ) )
45	44	eleq1d 2300	. . . . . 6 |- ( k e. NN -> ( ( ( ( k + 1 ) + 1 ) / 2 ) e. NN <-> ( ( k / 2 ) + 1 ) e. NN ) )
46	32, 45	imbitrrid 156	. . . . 5 |- ( k e. NN -> ( ( k / 2 ) e. NN -> ( ( ( k + 1 ) + 1 ) / 2 ) e. NN ) )
47	46	orim2d 795	. . . 4 |- ( k e. NN -> ( ( ( ( k + 1 ) / 2 ) e. NN \/ ( k / 2 ) e. NN ) -> ( ( ( k + 1 ) / 2 ) e. NN \/ ( ( ( k + 1 ) + 1 ) / 2 ) e. NN ) ) )
48		orcom 735	. . . 4 |- ( ( ( ( k + 1 ) / 2 ) e. NN \/ ( ( ( k + 1 ) + 1 ) / 2 ) e. NN ) <-> ( ( ( ( k + 1 ) + 1 ) / 2 ) e. NN \/ ( ( k + 1 ) / 2 ) e. NN ) )
49	47, 48	imbitrdi 161	. . 3 |- ( k e. NN -> ( ( ( ( k + 1 ) / 2 ) e. NN \/ ( k / 2 ) e. NN ) -> ( ( ( ( k + 1 ) + 1 ) / 2 ) e. NN \/ ( ( k + 1 ) / 2 ) e. NN ) ) )
50	6, 12, 18, 24, 31, 49	nnind 9158	. 2 |- ( N e. NN -> ( ( ( N + 1 ) / 2 ) e. NN \/ ( N / 2 ) e. NN ) )
51	50	orcomd 736	1 |- ( N e. NN -> ( ( N / 2 ) e. NN \/ ( ( N + 1 ) / 2 ) e. NN ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   + caddc 8034   # cap 8760   / cdiv 8851   NNcn 9142   2c2 9193

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201

This theorem is referenced by:  nneo  9582  zeo  9584