zeo - Intuitionistic Logic Explorer

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Theorem zeo 9317

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 9214	. 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
2		oveq1 5860	. . . . . 6 ⊢ (𝑁 = 0 → (𝑁 / 2) = (0 / 2))
3		2cn 8949	. . . . . . . 8 ⊢ 2 ∈ ℂ
4		2ap0 8971	. . . . . . . 8 ⊢ 2 # 0
5	3, 4	div0api 8663	. . . . . . 7 ⊢ (0 / 2) = 0
6		0z 9223	. . . . . . 7 ⊢ 0 ∈ ℤ
7	5, 6	eqeltri 2243	. . . . . 6 ⊢ (0 / 2) ∈ ℤ
8	2, 7	eqeltrdi 2261	. . . . 5 ⊢ (𝑁 = 0 → (𝑁 / 2) ∈ ℤ)
9	8	orcd 728	. . . 4 ⊢ (𝑁 = 0 → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
10	9	adantl 275	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝑁 = 0) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
11		nneoor 9314	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))
12		nnz 9231	. . . . . 6 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℤ)
13		nnz 9231	. . . . . 6 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ)
14	12, 13	orim12i 754	. . . . 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 275	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
17		nneoor 9314	. . . . 5 ⊢ (-𝑁 ∈ ℕ → ((-𝑁 / 2) ∈ ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ))
18	17	adantl 275	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → ((-𝑁 / 2) ∈ ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ))
19		recn 7907	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ)
20		divnegap 8623	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → -(𝑁 / 2) = (-𝑁 / 2))
21	3, 4, 20	mp3an23 1324	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℂ → -(𝑁 / 2) = (-𝑁 / 2))
22	19, 21	syl 14	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → -(𝑁 / 2) = (-𝑁 / 2))
23	22	eleq1d 2239	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (-(𝑁 / 2) ∈ ℕ ↔ (-𝑁 / 2) ∈ ℕ))
24		nnnegz 9215	. . . . . . . 8 ⊢ (-(𝑁 / 2) ∈ ℕ → --(𝑁 / 2) ∈ ℤ)
25	23, 24	syl6bir 163	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → ((-𝑁 / 2) ∈ ℕ → --(𝑁 / 2) ∈ ℤ))
26	19	halfcld 9122	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (𝑁 / 2) ∈ ℂ)
27	26	negnegd 8221	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → --(𝑁 / 2) = (𝑁 / 2))
28	27	eleq1d 2239	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (--(𝑁 / 2) ∈ ℤ ↔ (𝑁 / 2) ∈ ℤ))
29	25, 28	sylibd 148	. . . . . 6 ⊢ (𝑁 ∈ ℝ → ((-𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℤ))
30		nnz 9231	. . . . . . 7 ⊢ (((-𝑁 + 1) / 2) ∈ ℕ → ((-𝑁 + 1) / 2) ∈ ℤ)
31		peano2zm 9250	. . . . . . . . . 10 ⊢ (((-𝑁 + 1) / 2) ∈ ℤ → (((-𝑁 + 1) / 2) − 1) ∈ ℤ)
32		ax-1cn 7867	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
33	32, 3	negsubdi2i 8205	. . . . . . . . . . . . . . . . . 18 ⊢ -(1 − 2) = (2 − 1)
34		2m1e1 8996	. . . . . . . . . . . . . . . . . 18 ⊢ (2 − 1) = 1
35	33, 34	eqtr2i 2192	. . . . . . . . . . . . . . . . 17 ⊢ 1 = -(1 − 2)
36	32, 3	subcli 8195	. . . . . . . . . . . . . . . . . 18 ⊢ (1 − 2) ∈ ℂ
37	32, 36	negcon2i 8202	. . . . . . . . . . . . . . . . 17 ⊢ (1 = -(1 − 2) ↔ (1 − 2) = -1)
38	35, 37	mpbi 144	. . . . . . . . . . . . . . . 16 ⊢ (1 − 2) = -1
39	38	oveq2i 5864	. . . . . . . . . . . . . . 15 ⊢ (-𝑁 + (1 − 2)) = (-𝑁 + -1)
40		negcl 8119	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℂ → -𝑁 ∈ ℂ)
41		addsubass 8129	. . . . . . . . . . . . . . . . 17 ⊢ ((-𝑁 ∈ ℂ ∧ 1 ∈ ℂ ∧ 2 ∈ ℂ) → ((-𝑁 + 1) − 2) = (-𝑁 + (1 − 2)))
42	32, 3, 41	mp3an23 1324	. . . . . . . . . . . . . . . 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 8176	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑁 + 1) = (-𝑁 + -1))
45	32, 44	mpan2 423	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → -(𝑁 + 1) = (-𝑁 + -1))
46	39, 43, 45	3eqtr4a 2229	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = -(𝑁 + 1))
47	46	oveq1d 5868	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (-(𝑁 + 1) / 2))
48		2div2e1 9010	. . . . . . . . . . . . . . . 16 ⊢ (2 / 2) = 1
49	48	eqcomi 2174	. . . . . . . . . . . . . . 15 ⊢ 1 = (2 / 2)
50	49	oveq2i 5864	. . . . . . . . . . . . . 14 ⊢ (((-𝑁 + 1) / 2) − 1) = (((-𝑁 + 1) / 2) − (2 / 2))
51		peano2cn 8054	. . . . . . . . . . . . . . . 16 ⊢ (-𝑁 ∈ ℂ → (-𝑁 + 1) ∈ ℂ)
52	40, 51	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → (-𝑁 + 1) ∈ ℂ)
53	3, 4	pm3.2i 270	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ ℂ ∧ 2 # 0)
54		divsubdirap 8625	. . . . . . . . . . . . . . . 16 ⊢ (((-𝑁 + 1) ∈ ℂ ∧ 2 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0)) → (((-𝑁 + 1) − 2) / 2) = (((-𝑁 + 1) / 2) − (2 / 2)))
55	3, 53, 54	mp3an23 1324	. . . . . . . . . . . . . . 15 ⊢ ((-𝑁 + 1) ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (((-𝑁 + 1) / 2) − (2 / 2)))
56	52, 55	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (((-𝑁 + 1) / 2) − (2 / 2)))
57	50, 56	eqtr4id 2222	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) / 2) − 1) = (((-𝑁 + 1) − 2) / 2))
58		peano2cn 8054	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
59		divnegap 8623	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 + 1) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → -((𝑁 + 1) / 2) = (-(𝑁 + 1) / 2))
60	3, 4, 59	mp3an23 1324	. . . . . . . . . . . . . 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 2213	. . . . . . . . . . . 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 2239	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → ((((-𝑁 + 1) / 2) − 1) ∈ ℤ ↔ -((𝑁 + 1) / 2) ∈ ℤ))
65	31, 64	syl5ib 153	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ → -((𝑁 + 1) / 2) ∈ ℤ))
66		znegcl 9243	. . . . . . . . 9 ⊢ (-((𝑁 + 1) / 2) ∈ ℤ → --((𝑁 + 1) / 2) ∈ ℤ)
67	65, 66	syl6 33	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ → --((𝑁 + 1) / 2) ∈ ℤ))
68		peano2re 8055	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
69	68	recnd 7948	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℂ)
70	69	halfcld 9122	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → ((𝑁 + 1) / 2) ∈ ℂ)
71	70	negnegd 8221	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → --((𝑁 + 1) / 2) = ((𝑁 + 1) / 2))
72	71	eleq1d 2239	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (--((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 + 1) / 2) ∈ ℤ))
73	67, 72	sylibd 148	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ → ((𝑁 + 1) / 2) ∈ ℤ))
74	30, 73	syl5 32	. . . . . 6 ⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ))
75	29, 74	orim12d 781	. . . . 5 ⊢ (𝑁 ∈ ℝ → (((-𝑁 / 2) ∈ ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)))
76	75	adantr 274	. . . 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 1301	. 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
79	1, 78	sylbi 120	1 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∨ wo 703 ∨ w3o 972 = wceq 1348 ∈ wcel 2141 class class class wbr 3989 (class class class)co 5853 ℂcc 7772 ℝcr 7773 0cc0 7774 1c1 7775 + caddc 7777 − cmin 8090 -cneg 8091 # cap 8500 / cdiv 8589 ℕcn 8878 2c2 8929 ℤcz 9212

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892

This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213

This theorem is referenced by: zeo2 9318 zeo3 11827 mulsucdiv2z 11844 abssinper 13561

W3C validator