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Theorem nno 16446

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 12952	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≠ 1))
2		nnnn0 12517	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
3		nn0o1gt2 16445	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 = 1 ∨ 2 < 𝑁))
4	2, 3	sylan 591	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 = 1 ∨ 2 < 𝑁))
5		eqneqall 2968	. . . . . . 7 ⊢ (𝑁 = 1 → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ))
6	5	a1d 26	. . . . . 6 ⊢ (𝑁 = 1 → ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ)))
7		nn0z 12621	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) / 2) ∈ ℕ₀ → ((𝑁 + 1) / 2) ∈ ℤ)
8		peano2zm 12643	. . . . . . . . . . . 12 ⊢ (((𝑁 + 1) / 2) ∈ ℤ → (((𝑁 + 1) / 2) − 1) ∈ ℤ)
9	7, 8	syl 18	. . . . . . . . . . 11 ⊢ (((𝑁 + 1) / 2) ∈ ℕ₀ → (((𝑁 + 1) / 2) − 1) ∈ ℤ)
10	9	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → (((𝑁 + 1) / 2) − 1) ∈ ℤ)
11		2cn 12322	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℂ
12	11	mullidi 11220	. . . . . . . . . . . . . 14 ⊢ (1 · 2) = 2
13		nnre 12246	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
14	13	ltp1d 12151	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → 𝑁 < (𝑁 + 1))
15	14	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 𝑁 < (𝑁 + 1))
16		2re 12321	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
17		peano2nn 12251	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
18	17	nnred 12254	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ)
19		lttr 11292	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ) → ((2 < 𝑁 ∧ 𝑁 < (𝑁 + 1)) → 2 < (𝑁 + 1)))
20	16, 13, 18, 19	mp3an2i 1494	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → ((2 < 𝑁 ∧ 𝑁 < (𝑁 + 1)) → 2 < (𝑁 + 1)))
21	20	expdimp 457	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → (𝑁 < (𝑁 + 1) → 2 < (𝑁 + 1)))
22	15, 21	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 2 < (𝑁 + 1))
23	12, 22	eqbrtrid 5145	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → (1 · 2) < (𝑁 + 1))
24		1red 11215	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 1 ∈ ℝ)
25	18	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → (𝑁 + 1) ∈ ℝ)
26		2rp 13027	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ⁺
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 2 ∈ ℝ⁺)
28	24, 25, 27	ltmuldivd 13113	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → ((1 · 2) < (𝑁 + 1) ↔ 1 < ((𝑁 + 1) / 2)))
29	23, 28	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 1 < ((𝑁 + 1) / 2))
30	18	rehalfcld 12497	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) / 2) ∈ ℝ)
31	30	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → ((𝑁 + 1) / 2) ∈ ℝ)
32	24, 31	posdifd 11807	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → (1 < ((𝑁 + 1) / 2) ↔ 0 < (((𝑁 + 1) / 2) − 1)))
33	29, 32	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 2 < 𝑁) → 0 < (((𝑁 + 1) / 2) − 1))
34	33	adantlr 727	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → 0 < (((𝑁 + 1) / 2) − 1))
35		elnnz 12607	. . . . . . . . . 10 ⊢ ((((𝑁 + 1) / 2) − 1) ∈ ℕ ↔ ((((𝑁 + 1) / 2) − 1) ∈ ℤ ∧ 0 < (((𝑁 + 1) / 2) − 1)))
36	10, 34, 35	sylanbrc 594	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → (((𝑁 + 1) / 2) − 1) ∈ ℕ)
37		nncn 12247	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
38		xp1d2m1eqxm1d2 12504	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℂ → (((𝑁 + 1) / 2) − 1) = ((𝑁 − 1) / 2))
39	37, 38	syl 18	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑁 + 1) / 2) − 1) = ((𝑁 − 1) / 2))
40	39	eleq1d 2847	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → ((((𝑁 + 1) / 2) − 1) ∈ ℕ ↔ ((𝑁 − 1) / 2) ∈ ℕ))
41	40	adantr 485	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((((𝑁 + 1) / 2) − 1) ∈ ℕ ↔ ((𝑁 − 1) / 2) ∈ ℕ))
42	41	adantr 485	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → ((((𝑁 + 1) / 2) − 1) ∈ ℕ ↔ ((𝑁 − 1) / 2) ∈ ℕ))
43	36, 42	mpbid 235	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → ((𝑁 − 1) / 2) ∈ ℕ)
44	43	a1d 26	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) ∧ 2 < 𝑁) → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ))
45	44	expcom 418	. . . . . 6 ⊢ (2 < 𝑁 → ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ)))
46	6, 45	jaoi 870	. . . . 5 ⊢ ((𝑁 = 1 ∨ 2 < 𝑁) → ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ)))
47	4, 46	mpcom 39	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 ≠ 1 → ((𝑁 − 1) / 2) ∈ ℕ))
48	47	impancom 456	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≠ 1) → (((𝑁 + 1) / 2) ∈ ℕ₀ → ((𝑁 − 1) / 2) ∈ ℕ))
49	1, 48	sylbi 220	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (((𝑁 + 1) / 2) ∈ ℕ₀ → ((𝑁 − 1) / 2) ∈ ℕ))
50	49	imp 411	1 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 class class class wbr 5108 ‘cfv 6536 (class class class)co 7412 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11249 − cmin 11447 / cdiv 11877 ℕcn 12239 2c2 12301 ℕ₀cn0 12510 ℤcz 12597 ℤ_≥cuz 12868 ℝ⁺crp 13022

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-er 8692 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-n0 12511 df-z 12598 df-uz 12869 df-rp 13023

This theorem is used by: nn0o 16447 gausslemma2dlem0b 27532 blennngt2o2 49400 dignn0flhalf 49426

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