Theorem nneo 6144

Description: A natural number is even or odd but not both.

Hypothesis

Ref	Expression
nneo.1	|- N e. NN

Assertion

Ref	Expression
nneo	|- ((N / 2) e. NN <-> -. ((N + 1) / 2) e. NN)

Proof of Theorem nneo

Step	Hyp	Ref	Expression
1		1lt2 5975	. . . . . . . 8 |- 1 < 2
2		1re 5407	. . . . . . . . 9 |- 1 e. RR
3		2re 5926	. . . . . . . . 9 |- 2 e. RR
4		nneo.1	. . . . . . . . . 10 |- N e. NN
5	4	nnre 5879	. . . . . . . . 9 |- N e. RR
6	2, 3, 5	ltadd2 5564	. . . . . . . 8 |- (1 < 2 <-> (N + 1) < (N + 2))
7	1, 6	mpbi 189	. . . . . . 7 |- (N + 1) < (N + 2)
8		2cn 5927	. . . . . . . 8 |- 2 e. CC
9	5, 2	readdcl 5306	. . . . . . . . 9 |- (N + 1) e. RR
10	9	recn 5286	. . . . . . . 8 |- (N + 1) e. CC
11		2ne0 5937	. . . . . . . 8 |- 2 =/= 0
12	8, 10, 11	divcan2 5685	. . . . . . 7 |- (2 x. ((N + 1) / 2)) = (N + 1)
13	5, 3, 11	redivcl 5754	. . . . . . . . . 10 |- (N / 2) e. RR
14	13	recn 5286	. . . . . . . . 9 |- (N / 2) e. CC
15		ax1cn 5241	. . . . . . . . 9 |- 1 e. CC
16	8, 14, 15	adddi 5298	. . . . . . . 8 |- (2 x. ((N / 2) + 1)) = ((2 x. (N / 2)) + (2 x. 1))
17	4	nncn 5880	. . . . . . . . . 10 |- N e. CC
18	8, 17, 11	divcan2 5685	. . . . . . . . 9 |- (2 x. (N / 2)) = N
19	8	mulid1 5304	. . . . . . . . 9 |- (2 x. 1) = 2
20	18, 19	opreq12i 3958	. . . . . . . 8 |- ((2 x. (N / 2)) + (2 x. 1)) = (N + 2)
21	16, 20	eqtr 1487	. . . . . . 7 |- (2 x. ((N / 2) + 1)) = (N + 2)
22	7, 12, 21	3brtr4 2633	. . . . . 6 |- (2 x. ((N + 1) / 2)) < (2 x. ((N / 2) + 1))
23		2pos 5936	. . . . . . 7 |- 0 < 2
24	9, 3, 11	redivcl 5754	. . . . . . . 8 |- ((N + 1) / 2) e. RR
25	13, 2	readdcl 5306	. . . . . . . 8 |- ((N / 2) + 1) e. RR
26	24, 25, 3	ltmul2 5790	. . . . . . 7 |- (0 < 2 -> (((N + 1) / 2) < ((N / 2) + 1) <-> (2 x. ((N + 1) / 2)) < (2 x. ((N / 2) + 1))))
27	23, 26	ax-mp 7	. . . . . 6 |- (((N + 1) / 2) < ((N / 2) + 1) <-> (2 x. ((N + 1) / 2)) < (2 x. ((N / 2) + 1)))
28	22, 27	mpbir 190	. . . . 5 |- ((N + 1) / 2) < ((N / 2) + 1)
29	24, 25	ltnle 5552	. . . . 5 |- (((N + 1) / 2) < ((N / 2) + 1) <-> -. ((N / 2) + 1) <_ ((N + 1) / 2))
30	28, 29	mpbi 189	. . . 4 |- -. ((N / 2) + 1) <_ ((N + 1) / 2)
31	5	ltp1 5769	. . . . . 6 |- N < (N + 1)
32	5, 9, 3, 23	ltdiv1i 5779	. . . . . 6 |- (N < (N + 1) <-> (N / 2) < ((N + 1) / 2))
33	31, 32	mpbi 189	. . . . 5 |- (N / 2) < ((N + 1) / 2)
34		nnltp1let 5902	. . . . 5 |- (((N / 2) e. NN /\ ((N + 1) / 2) e. NN) -> ((N / 2) < ((N + 1) / 2) <-> ((N / 2) + 1) <_ ((N + 1) / 2)))
35	33, 34	mpbii 193	. . . 4 |- (((N / 2) e. NN /\ ((N + 1) / 2) e. NN) -> ((N / 2) + 1) <_ ((N + 1) / 2))
36	30, 35	mto 106	. . 3 |- -. ((N / 2) e. NN /\ ((N + 1) / 2) e. NN)
37		imnan 242	. . 3 |- (((N / 2) e. NN -> -. ((N + 1) / 2) e. NN) <-> -. ((N / 2) e. NN /\ ((N + 1) / 2) e. NN))
38	36, 37	mpbir 190	. 2 |- ((N / 2) e. NN -> -. ((N + 1) / 2) e. NN)
39		opreq1 3953	. . . . . . . 8 |- (j = 1 -> (j + 1) = (1 + 1))
40	39	opreq1d 3960	. . . . . . 7 |- (j = 1 -> ((j + 1) / 2) = ((1 + 1) / 2))
41	40	eleq1d 1532	. . . . . 6 |- (j = 1 -> (((j + 1) / 2) e. NN <-> ((1 + 1) / 2) e. NN))
42		opreq1 3953	. . . . . . 7 |- (j = 1 -> (j / 2) = (1 / 2))
43	42	eleq1d 1532	. . . . . 6 |- (j = 1 -> ((j / 2) e. NN <-> (1 / 2) e. NN))
44	41, 43	orbi12d 625	. . . . 5 |- (j = 1 -> ((((j + 1) / 2) e. NN \/ (j / 2) e. NN) <-> (((1 + 1) / 2) e. NN \/ (1 / 2) e. NN)))
45		opreq1 3953	. . . . . . . 8 |- (j = k -> (j + 1) = (k + 1))
46	45	opreq1d 3960	. . . . . . 7 |- (j = k -> ((j + 1) / 2) = ((k + 1) / 2))
47	46	eleq1d 1532	. . . . . 6 |- (j = k -> (((j + 1) / 2) e. NN <-> ((k + 1) / 2) e. NN))
48		opreq1 3953	. . . . . . 7 |- (j = k -> (j / 2) = (k / 2))
49	48	eleq1d 1532	. . . . . 6 |- (j = k -> ((j / 2) e. NN <-> (k / 2) e. NN))
50	47, 49	orbi12d 625	. . . . 5 |- (j = k -> ((((j + 1) / 2) e. NN \/ (j / 2) e. NN) <-> (((k + 1) / 2) e. NN \/ (k / 2) e. NN)))
51		opreq1 3953	. . . . . . . 8 |- (j = (k + 1) -> (j + 1) = ((k + 1) + 1))
52	51	opreq1d 3960	. . . . . . 7 |- (j = (k + 1) -> ((j + 1) / 2) = (((k + 1) + 1) / 2))
53	52	eleq1d 1532	. . . . . 6 |- (j = (k + 1) -> (((j + 1) / 2) e. NN <-> (((k + 1) + 1) / 2) e. NN))
54		opreq1 3953	. . . . . . 7 |- (j = (k + 1) -> (j / 2) = ((k + 1) / 2))
55	54	eleq1d 1532	. . . . . 6 |- (j = (k + 1) -> ((j / 2) e. NN <-> ((k + 1) / 2) e. NN))
56	53, 55	orbi12d 625	. . . . 5 |- (j = (k + 1) -> ((((j + 1) / 2) e. NN \/ (j / 2) e. NN) <-> ((((k + 1) + 1) / 2) e. NN \/ ((k + 1) / 2) e. NN)))
57		opreq1 3953	. . . . . . . 8 |- (j = N -> (j + 1) = (N + 1))
58	57	opreq1d 3960	. . . . . . 7 |- (j = N -> ((j + 1) / 2) = ((N + 1) / 2))
59	58	eleq1d 1532	. . . . . 6 |- (j = N -> (((j + 1) / 2) e. NN <-> ((N + 1) / 2) e. NN))
60		opreq1 3953	. . . . . . 7 |- (j = N -> (j / 2) = (N / 2))
61	60	eleq1d 1532	. . . . . 6 |- (j = N -> ((j / 2) e. NN <-> (N / 2) e. NN))
62	59, 61	orbi12d 625	. . . . 5 |- (j = N -> ((((j + 1) / 2) e. NN \/ (j / 2) e. NN) <-> (((N + 1) / 2) e. NN \/ (N / 2) e. NN)))
63		df-2 5917	. . . . . . . . 9 |- 2 = (1 + 1)
64	63	opreq1i 3956	. . . . . . . 8 |- (2 / 2) = ((1 + 1) / 2)
65	8, 11	divid 5726	. . . . . . . 8 |- (2 / 2) = 1
66	64, 65	eqtr3 1489	. . . . . . 7 |- ((1 + 1) / 2) = 1
67		1nn 5882	. . . . . . 7 |- 1 e. NN
68	66, 67	eqeltr 1536	. . . . . 6 |- ((1 + 1) / 2) e. NN
69	68	orci 270	. . . . 5 |- (((1 + 1) / 2) e. NN \/ (1 / 2) e. NN)
70		nncnt 5878	. . . . . . . . . 10 |- (k e. NN -> k e. CC)
71		axaddass 5249	. . . . . . . . . . . . . 14 |- ((k e. CC /\ 1 e. CC /\ 1 e. CC) -> ((k + 1) + 1) = (k + (1 + 1)))
72	15, 15, 71	mp3an23 905	. . . . . . . . . . . . 13 |- (k e. CC -> ((k + 1) + 1) = (k + (1 + 1)))
73	63	opreq2i 3957	. . . . . . . . . . . . 13 |- (k + 2) = (k + (1 + 1))
74	72, 73	syl6eqr 1517	. . . . . . . . . . . 12 |- (k e. CC -> ((k + 1) + 1) = (k + 2))
75	74	opreq1d 3960	. . . . . . . . . . 11 |- (k e. CC -> (((k + 1) + 1) / 2) = ((k + 2) / 2))
76		divdirt 5713	. . . . . . . . . . . . . 14 |- (((k e. CC /\ 2 e. CC /\ 2 e. CC) /\ 2 =/= 0) -> ((k + 2) / 2) = ((k / 2) + (2 / 2)))
77	11, 76	mpan2 694	. . . . . . . . . . . . 13 |- ((k e. CC /\ 2 e. CC /\ 2 e. CC) -> ((k + 2) / 2) = ((k / 2) + (2 / 2)))
78	8, 8, 77	mp3an23 905	. . . . . . . . . . . 12 |- (k e. CC -> ((k + 2) / 2) = ((k / 2) + (2 / 2)))
79	65	opreq2i 3957	. . . . . . . . . . . 12 |- ((k / 2) + (2 / 2)) = ((k / 2) + 1)
80	78, 79	syl6eq 1515	. . . . . . . . . . 11 |- (k e. CC -> ((k + 2) / 2) = ((k / 2) + 1))
81	75, 80	eqtrd 1499	. . . . . . . . . 10 |- (k e. CC -> (((k + 1) + 1) / 2) = ((k / 2) + 1))
82	70, 81	syl 10	. . . . . . . . 9 |- (k e. NN -> (((k + 1) + 1) / 2) = ((k / 2) + 1))
83	82	eleq1d 1532	. . . . . . . 8 |- (k e. NN -> ((((k + 1) + 1) / 2) e. NN <->