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Theorem nno 16019

Description: An alternate characterization of an odd integer greater than 1. (Contributed by AV, 2-Jun-2020.) (Proof shortened by AV, 10-Jul-2022.)

**Assertion**
Ref	Expression
nno	⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ)

**Proof of Theorem nno**
Step	Hyp	Ref	Expression
1		eluz2b3 12591	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
2		nnnn0 12170	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3		nn0o1gt2 16018	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 = 1 ∨ 2 < 𝑁))
4	2, 3	sylan 579	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 = 1 ∨ 2 < 𝑁))
5		eqneqall 2953	. . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ))
6	5	a1d 25	. . . . . 6 ⊢ (𝑁 = 1 → ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ)))
7		nn0z 12273	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) / 2) ∈ ℕ₀ → ((𝑁 + 1) / 2) ∈ ℤ)
8		peano2zm 12293	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) / 2) ∈ ℤ → (((𝑁 + 1) / 2) − 1) ∈ ℤ)
9	7, 8	syl 17	. . . . . . . . . . 11 ⊢ (((𝑁 + 1) / 2) ∈ ℕ₀ → (((𝑁 + 1) / 2) − 1) ∈ ℤ)
10	9	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → (((𝑁 + 1) / 2) − 1) ∈ ℤ)
11		2cn 11978	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
12	11	mulid2i 10911	. . . . . . . . . . . . . 14 ⊢ (1 · 2) = 2
13		nnre 11910	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
14	13	ltp1d 11835	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → 𝑁 < (𝑁 + 1))
15	14	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 𝑁 < (𝑁 + 1))
16		2re 11977	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
17		peano2nn 11915	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
18	17	nnred 11918	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ)
19		lttr 10982	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ) → ((2 < 𝑁 ∧ 𝑁 < (𝑁 + 1)) → 2 < (𝑁 + 1)))
20	16, 13, 18, 19	mp3an2i 1464	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → ((2 < 𝑁 ∧ 𝑁 < (𝑁 + 1)) → 2 < (𝑁 + 1)))
21	20	expdimp 452	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → (𝑁 < (𝑁 + 1) → 2 < (𝑁 + 1)))
22	15, 21	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 2 < (𝑁 + 1))
23	12, 22	eqbrtrid 5105	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → (1 · 2) < (𝑁 + 1))
24		1red 10907	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 1 ∈ ℝ)
25	18	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → (𝑁 + 1) ∈ ℝ)
26		2rp 12664	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ⁺
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 2 ∈ ℝ⁺)
28	24, 25, 27	ltmuldivd 12748	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → ((1 · 2) < (𝑁 + 1) ↔ 1 < ((𝑁 + 1) / 2)))
29	23, 28	mpbid 231	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 1 < ((𝑁 + 1) / 2))
30	18	rehalfcld 12150	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℝ)
31	30	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → ((𝑁 + 1) / 2) ∈ ℝ)
32	24, 31	posdifd 11492	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → (1 < ((𝑁 + 1) / 2) ↔ 0 < (((𝑁 + 1) / 2) − 1)))
33	29, 32	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 0 < (((𝑁 + 1) / 2) − 1))
34	33	adantlr 711	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → 0 < (((𝑁 + 1) / 2) − 1))
35		elnnz 12259	. . . . . . . . . 10 ⊢ ((((𝑁 + 1) / 2) − 1) ∈ ℕ ↔ ((((𝑁 + 1) / 2) − 1) ∈ ℤ ∧ 0 < (((𝑁 + 1) / 2) − 1)))
36	10, 34, 35	sylanbrc 582	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → (((𝑁 + 1) / 2) − 1) ∈ ℕ)
37		nncn 11911	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
38		xp1d2m1eqxm1d2 12157	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (((𝑁 + 1) / 2) − 1) = ((𝑁 − 1) / 2))
39	37, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) − 1) = ((𝑁 − 1) / 2))
40	39	eleq1d 2823	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((((𝑁 + 1) / 2) − 1) ∈ ℕ ↔ ((𝑁 − 1) / 2) ∈ ℕ))
41	40	adantr 480	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((((𝑁 + 1) / 2) − 1) ∈ ℕ ↔ ((𝑁 − 1) / 2) ∈ ℕ))
42	41	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → ((((𝑁 + 1) / 2) − 1) ∈ ℕ ↔ ((𝑁 − 1) / 2) ∈ ℕ))
43	36, 42	mpbid 231	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → ((𝑁 − 1) / 2) ∈ ℕ)
44	43	a1d 25	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ))
45	44	expcom 413	. . . . . 6 ⊢ (2 < 𝑁 → ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ)))
46	6, 45	jaoi 853	. . . . 5 ⊢ ((𝑁 = 1 ∨ 2 < 𝑁) → ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ)))
47	4, 46	mpcom 38	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ))
48	47	impancom 451	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≠ 1) → (((𝑁 + 1) / 2) ∈ ℕ₀ → ((𝑁 − 1) / 2) ∈ ℕ))
49	1, 48	sylbi 216	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (((𝑁 + 1) / 2) ∈ ℕ₀ → ((𝑁 − 1) / 2) ∈ ℕ))
50	49	imp 406	1 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 − cmin 11135 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660

This theorem is referenced by: nn0o 16020 gausslemma2dlem0b 26410 blennngt2o2 45826 dignn0flhalf 45852

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