zeo - Intuitionistic Logic Explorer

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Theorem zeo 9683

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 9579	. 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
2		oveq1 6057	. . . . . 6 ⊢ (𝑁 = 0 → (𝑁 / 2) = (0 / 2))
3		2cn 9308	. . . . . . . 8 ⊢ 2 ∈ ℂ
4		2ap0 9330	. . . . . . . 8 ⊢ 2 # 0
5	3, 4	div0api 9020	. . . . . . 7 ⊢ (0 / 2) = 0
6		0z 9588	. . . . . . 7 ⊢ 0 ∈ ℤ
7	5, 6	eqeltri 2305	. . . . . 6 ⊢ (0 / 2) ∈ ℤ
8	2, 7	eqeltrdi 2323	. . . . 5 ⊢ (𝑁 = 0 → (𝑁 / 2) ∈ ℤ)
9	8	orcd 741	. . . 4 ⊢ (𝑁 = 0 → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
10	9	adantl 277	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝑁 = 0) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
11		nneoor 9680	. . . . 5 ⊢ (𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))
12		nnz 9596	. . . . . 6 ⊢ ((𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℤ)
13		nnz 9596	. . . . . 6 ⊢ (((𝑁 + 1) / 2) ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℤ)
14	12, 13	orim12i 767	. . . . 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 277	. . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
17		nneoor 9680	. . . . 5 ⊢ (-𝑁 ∈ ℕ → ((-𝑁 / 2) ∈ ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ))
18	17	adantl 277	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → ((-𝑁 / 2) ∈ ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ))
19		recn 8260	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ)
20		divnegap 8980	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → -(𝑁 / 2) = (-𝑁 / 2))
21	3, 4, 20	mp3an23 1366	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℂ → -(𝑁 / 2) = (-𝑁 / 2))
22	19, 21	syl 14	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → -(𝑁 / 2) = (-𝑁 / 2))
23	22	eleq1d 2301	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (-(𝑁 / 2) ∈ ℕ ↔ (-𝑁 / 2) ∈ ℕ))
24		nnnegz 9580	. . . . . . . 8 ⊢ (-(𝑁 / 2) ∈ ℕ → --(𝑁 / 2) ∈ ℤ)
25	23, 24	biimtrrdi 164	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → ((-𝑁 / 2) ∈ ℕ → --(𝑁 / 2) ∈ ℤ))
26	19	halfcld 9483	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (𝑁 / 2) ∈ ℂ)
27	26	negnegd 8575	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → --(𝑁 / 2) = (𝑁 / 2))
28	27	eleq1d 2301	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (--(𝑁 / 2) ∈ ℤ ↔ (𝑁 / 2) ∈ ℤ))
29	25, 28	sylibd 149	. . . . . 6 ⊢ (𝑁 ∈ ℝ → ((-𝑁 / 2) ∈ ℕ → (𝑁 / 2) ∈ ℤ))
30		nnz 9596	. . . . . . 7 ⊢ (((-𝑁 + 1) / 2) ∈ ℕ → ((-𝑁 + 1) / 2) ∈ ℤ)
31		peano2zm 9615	. . . . . . . . . 10 ⊢ (((-𝑁 + 1) / 2) ∈ ℤ → (((-𝑁 + 1) / 2) − 1) ∈ ℤ)
32		ax-1cn 8220	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
33	32, 3	negsubdi2i 8559	. . . . . . . . . . . . . . . . . 18 ⊢ -(1 − 2) = (2 − 1)
34		2m1e1 9355	. . . . . . . . . . . . . . . . . 18 ⊢ (2 − 1) = 1
35	33, 34	eqtr2i 2254	. . . . . . . . . . . . . . . . 17 ⊢ 1 = -(1 − 2)
36	32, 3	subcli 8549	. . . . . . . . . . . . . . . . . 18 ⊢ (1 − 2) ∈ ℂ
37	32, 36	negcon2i 8556	. . . . . . . . . . . . . . . . 17 ⊢ (1 = -(1 − 2) ↔ (1 − 2) = -1)
38	35, 37	mpbi 145	. . . . . . . . . . . . . . . 16 ⊢ (1 − 2) = -1
39	38	oveq2i 6061	. . . . . . . . . . . . . . 15 ⊢ (-𝑁 + (1 − 2)) = (-𝑁 + -1)
40		negcl 8473	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℂ → -𝑁 ∈ ℂ)
41		addsubass 8483	. . . . . . . . . . . . . . . . 17 ⊢ ((-𝑁 ∈ ℂ ∧ 1 ∈ ℂ ∧ 2 ∈ ℂ) → ((-𝑁 + 1) − 2) = (-𝑁 + (1 − 2)))
42	32, 3, 41	mp3an23 1366	. . . . . . . . . . . . . . . 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 8530	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → -(𝑁 + 1) = (-𝑁 + -1))
45	32, 44	mpan2 425	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → -(𝑁 + 1) = (-𝑁 + -1))
46	39, 43, 45	3eqtr4a 2291	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → ((-𝑁 + 1) − 2) = -(𝑁 + 1))
47	46	oveq1d 6065	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) − 2) / 2) = (-(𝑁 + 1) / 2))
48		2div2e1 9370	. . . . . . . . . . . . . . . 16 ⊢ (2 / 2) = 1
49	48	eqcomi 2236	. . . . . . . . . . . . . . 15 ⊢ 1 = (2 / 2)
50	49	oveq2i 6061	. . . . . . . . . . . . . 14 ⊢ (((-𝑁 + 1) / 2) − 1) = (((-𝑁 + 1) / 2) − (2 / 2))
51		peano2cn 8408	. . . . . . . . . . . . . . . 16 ⊢ (-𝑁 ∈ ℂ → (-𝑁 + 1) ∈ ℂ)
52	40, 51	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℂ → (-𝑁 + 1) ∈ ℂ)
53	3, 4	pm3.2i 272	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ ℂ ∧ 2 # 0)
54		divsubdirap 8982	. . . . . . . . . . . . . . . 16 ⊢ (((-𝑁 + 1) ∈ ℂ ∧ 2 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0)) → (((-𝑁 + 1) − 2) / 2) = (((-𝑁 + 1) / 2) − (2 / 2)))
55	3, 53, 54	mp3an23 1366	. . . . . . . . . . . . . . 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 2284	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (((-𝑁 + 1) / 2) − 1) = (((-𝑁 + 1) − 2) / 2))
58		peano2cn 8408	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℂ → (𝑁 + 1) ∈ ℂ)
59		divnegap 8980	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 + 1) ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 # 0) → -((𝑁 + 1) / 2) = (-(𝑁 + 1) / 2))
60	3, 4, 59	mp3an23 1366	. . . . . . . . . . . . . 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 2275	. . . . . . . . . . . 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 2301	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → ((((-𝑁 + 1) / 2) − 1) ∈ ℤ ↔ -((𝑁 + 1) / 2) ∈ ℤ))
65	31, 64	imbitrid 154	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → (((-𝑁 + 1) / 2) ∈ ℤ → -((𝑁 + 1) / 2) ∈ ℤ))
66		znegcl 9608	. . . . . . . . 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 8409	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
69	68	recnd 8302	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℂ)
70	69	halfcld 9483	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℝ → ((𝑁 + 1) / 2) ∈ ℂ)
71	70	negnegd 8575	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → --((𝑁 + 1) / 2) = ((𝑁 + 1) / 2))
72	71	eleq1d 2301	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (--((𝑁 + 1) / 2) ∈ ℤ ↔ ((𝑁 + 1) / 2) ∈ ℤ))
73	67, 72	sylibd 149	. . . . . . 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 794	. . . . 5 ⊢ (𝑁 ∈ ℝ → (((-𝑁 / 2) ∈ ℕ ∨ ((-𝑁 + 1) / 2) ∈ ℕ) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ)))
76	75	adantr 276	. . . 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 1343	. 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))
79	1, 78	sylbi 121	1 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 2) ∈ ℤ ∨ ((𝑁 + 1) / 2) ∈ ℤ))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 716 ∨ w3o 1004 = wceq 1398 ∈ wcel 2203 class class class wbr 4109 (class class class)co 6050 ℂcc 8125 ℝcr 8126 0cc0 8127 1c1 8128 + caddc 8130 − cmin 8444 -cneg 8445 # cap 8855 / cdiv 8946 ℕcn 9237 2c2 9288 ℤcz 9577

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-br 4110 df-opab 4172 df-id 4414 df-po 4417 df-iso 4418 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-iota 5312 df-fun 5354 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-n0 9497 df-z 9578

This theorem is referenced by: zeo2 9684 zeo3 12554 mulsucdiv2z 12571 abssinper 15711 lgseisenlem1 15943

W3C validator