zeo - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > zeo		GIF version

Theorem zeo 8343

Description: An integer is even or odd. (Contributed by NM, 1-Jan-2006.)

**Assertion**
Ref	Expression
zeo	⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))

**Proof of Theorem zeo**
Step	Hyp	Ref	Expression
1		elz 8247	. 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
2		oveq1 5519	. . . . . 6 ⊢ (𝑁 = 0 → (𝑁 / 2) = (0 / 2))
3		2cn 7986	. . . . . . . 8 ⊢ 2 ∈ ℂ
4		2ap0 8009	. . . . . . . 8 ⊢ 2 # 0
5	3, 4	div0api 7722	. . . . . . 7 ⊢ (0 / 2) = 0
6		0z 8256	. . . . . . 7 ⊢ 0 ∈ ℤ
7	5, 6	eqeltri 2110	. . . . . 6 ⊢ (0 / 2) ∈ ℤ
8	2, 7	syl6eqel 2128	. . . . 5 ⊢ (𝑁 = 0 → (𝑁 / 2) ∈ ℤ)
9	8	orcd 652	. . . 4 ⊢ (𝑁 = 0 → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
10	9	adantl 262	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝑁 = 0) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
11		nneoor 8340	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))
12		nnz 8264	. . . . . 6 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℤ)
13		nnz 8264	. . . . . 6 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ)
14	12, 13	orim12i 676	. . . . 5 ⊢ (((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
15	11, 14	syl 14	. . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
16	15	adantl 262	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
17		nneoor 8340	. . . . 5 ⊢ (-𝑁 ∈ ℕ → ((-𝑁 / 2) ∈ ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ))
18	17	adantl 262	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → ((-𝑁 / 2) ∈ ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ))
19		recn 7014	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ)
20		divnegap 7683	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → -(𝑁 / 2) = (-𝑁 / 2))
21	3, 4, 20	mp3an23 1224	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℂ → -(𝑁 / 2) = (-𝑁 / 2))
22	19, 21	syl 14	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → -(𝑁 / 2) = (-𝑁 / 2))
23	22	eleq1d 2106	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (-(𝑁 / 2) ∈ ℕ ↔ (-𝑁 / 2) ∈ ℕ))
24		nnnegz 8248	. . . . . . . 8 ⊢ (-(𝑁 / 2) ∈ ℕ → --(𝑁 / 2) ∈ ℤ)
25	23, 24	syl6bir 153	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → ((-𝑁 / 2) ∈ ℕ → --(𝑁 / 2) ∈ ℤ))
26	19	halfcld 8169	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (𝑁 / 2) ∈ ℂ)
27	26	negnegd 7313	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → --(𝑁 / 2) = (𝑁 / 2))
28	27	eleq1d 2106	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (--(𝑁 / 2) ∈ ℤ ↔ (𝑁 / 2) ∈ ℤ))
29	25, 28	sylibd 138	. . . . . 6 ⊢ (𝑁 ∈ ℝ → ((-𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℤ))
30		nnz 8264	. . . . . . 7 ⊢ (((-𝑁 + 1) / 2) ∈ ℕ → ((-𝑁 + 1) / 2) ∈ ℤ)
31		peano2zm 8283	. . . . . . . . . 10 ⊢ (((-𝑁 + 1) / 2) ∈ ℤ → (((-𝑁 + 1) / 2) − 1) ∈ ℤ)
32		ax-1cn 6977	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
33	32, 3	negsubdi2i 7297	. . . . . . . . . . . . . . . . . 18 ⊢ -(1 − 2) = (2 − 1)
34		2m1e1 8034	. . . . . . . . . . . . . . . . . 18 ⊢ (2 − 1) = 1
35	33, 34	eqtr2i 2061	. . . . . . . . . . . . . . . . 17 ⊢ 1 = -(1 − 2)
36	32, 3	subcli 7287	. . . . . . . . . . . . . . . . . 18 ⊢ (1 − 2) ∈ ℂ
37	32, 36	negcon2i 7294	. . . . . . . . . . . . . . . . 17 ⊢ (1 = -(1 − 2) ↔ (1 − 2) = -1)
38	35, 37	mpbi 133	. . . . . . . . . . . . . . . 16 ⊢ (1 − 2) = -1
39	38	oveq2i 5523	. . . . . . . . . . . . . . 15 ⊢ (-𝑁 + (1 − 2)) = (-𝑁 + -1)
40		negcl 7211	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℂ → -𝑁 ∈ ℂ)
41		addsubass 7221	. . . . . . . . . . . . . . . . 17 ⊢ ((-𝑁 ∈ ℂ ∧ 1 ∈ ℂ ∧ 2 ∈ ℂ) → ((-𝑁 + 1) − 2) = (-𝑁 + (1 − 2)))
42	32, 3, 41	mp3an23 1224	. . . . . . . . . . . . . . . 16 ⊢ (-𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = (-𝑁 + (1 − 2)))
43	40, 42	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = (-𝑁 + (1 − 2)))
44		negdi 7268	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑁 + 1) = (-𝑁 + -1))
45	32, 44	mpan2 401	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → -(𝑁 + 1) = (-𝑁 + -1))
46	39, 43, 45	3eqtr4a 2098	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = -(𝑁 + 1))
47	46	oveq1d 5527	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (-(𝑁 + 1) / 2))
48		peano2cn 7148	. . . . . . . . . . . . . . . 16 ⊢ (-𝑁 ∈ ℂ → (-𝑁 + 1) ∈ ℂ)
49	40, 48	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → (-𝑁 + 1) ∈ ℂ)
50	3, 4	pm3.2i 257	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ ℂ ∧ 2 # 0)
51		divsubdirap 7684	. . . . . . . . . . . . . . . 16 ⊢ (((-𝑁 + 1) ∈ ℂ ∧ 2 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0)) → (((-𝑁 + 1) − 2) / 2) = (((-𝑁 + 1) / 2) − (2 / 2)))
52	3, 50, 51	mp3an23 1224	. . . . . . . . . . . . . . 15 ⊢ ((-𝑁 + 1) ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (((-𝑁 + 1) / 2) − (2 / 2)))
53	49, 52	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (((-𝑁 + 1) / 2) − (2 / 2)))
54		2div2e1 8042	. . . . . . . . . . . . . . . 16 ⊢ (2 / 2) = 1
55	54	eqcomi 2044	. . . . . . . . . . . . . . 15 ⊢ 1 = (2 / 2)
56	55	oveq2i 5523	. . . . . . . . . . . . . 14 ⊢ (((-𝑁 + 1) / 2) − 1) = (((-𝑁 + 1) / 2) − (2 / 2))
57	53, 56	syl6reqr 2091	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) / 2) − 1) = (((-𝑁 + 1) − 2) / 2))
58		peano2cn 7148	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
59		divnegap 7683	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 + 1) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → -((𝑁 + 1) / 2) = (-(𝑁 + 1) / 2))
60	3, 4, 59	mp3an23 1224	. . . . . . . . . . . . . 14 ⊢ ((𝑁 + 1) ∈ ℂ → -((𝑁 + 1) / 2) = (-(𝑁 + 1) / 2))
61	58, 60	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → -((𝑁 + 1) / 2) = (-(𝑁 + 1) / 2))
62	47, 57, 61	3eqtr4d 2082	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) / 2) − 1) = -((𝑁 + 1) / 2))
63	19, 62	syl 14	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) − 1) = -((𝑁 + 1) / 2))
64	63	eleq1d 2106	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → ((((-𝑁 + 1) / 2) − 1) ∈ ℤ ↔ -((𝑁 + 1) / 2) ∈ ℤ))
65	31, 64	syl5ib 143	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ → -((𝑁 + 1) / 2) ∈ ℤ))
66		znegcl 8276	. . . . . . . . 9 ⊢ (-((𝑁 + 1) / 2) ∈ ℤ → --((𝑁 + 1) / 2) ∈ ℤ)
67	65, 66	syl6 29	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ → --((𝑁 + 1) / 2) ∈ ℤ))
68		peano2re 7149	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
69	68	recnd 7054	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℂ)
70	69	halfcld 8169	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → ((𝑁 + 1) / 2) ∈ ℂ)
71	70	negnegd 7313	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → --((𝑁 + 1) / 2) = ((𝑁 + 1) / 2))
72	71	eleq1d 2106	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (--((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 + 1) / 2) ∈ ℤ))
73	67, 72	sylibd 138	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℤ))
74	30, 73	syl5 28	. . . . . 6 ⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ))
75	29, 74	orim12d 700	. . . . 5 ⊢ (𝑁 ∈ ℝ → (((-𝑁 / 2) ∈ ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)))
76	75	adantr 261	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (((-𝑁 / 2) ∈ ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)))
77	18, 76	mpd 13	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
78	10, 16, 77	3jaodan 1201	. 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
79	1, 78	sylbi 114	1 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ∨ wo 629 ∨ w3o 884 = wceq 1243 ∈ wcel 1393 class class class wbr 3764 (class class class)co 5512 ℂcc 6887 ℝcr 6888 0cc0 6889 1c1 6890 + caddc 6892 − cmin 7182 -cneg 7183 # cap 7572 / cdiv 7651 ℕcn 7914 2c2 7964 ℤcz 8245

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 ax-pre-mulext 7002

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rmo 2314 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-reap 7566 df-ap 7573 df-div 7652 df-inn 7915 df-2 7973 df-n0 8182 df-z 8246

This theorem is referenced by: zeo2 8344

W3C validator