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Theorem nn0o 16367

Description: An alternate characterization of an odd nonnegative integer. (Contributed by AV, 28-May-2020.) (Proof shortened by AV, 2-Jun-2020.)

**Assertion**
Ref	Expression
nn0o	⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ₀)

**Proof of Theorem nn0o**
Step	Hyp	Ref	Expression
1		nn0o1gt2 16365	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (𝑁 = 1 ∨ 2 < 𝑁))
2		1m1e0 12322	. . . . . . . 8 ⊢ (1 − 1) = 0
3	2	oveq1i 7436	. . . . . . 7 ⊢ ((1 − 1) / 2) = (0 / 2)
4		2cn 12325	. . . . . . . 8 ⊢ 2 ∈ ℂ
5		2ne0 12354	. . . . . . . 8 ⊢ 2 ≠ 0
6	4, 5	div0i 11986	. . . . . . 7 ⊢ (0 / 2) = 0
7	3, 6	eqtri 2756	. . . . . 6 ⊢ ((1 − 1) / 2) = 0
8		0nn0 12525	. . . . . 6 ⊢ 0 ∈ ℕ₀
9	7, 8	eqeltri 2825	. . . . 5 ⊢ ((1 − 1) / 2) ∈ ℕ₀
10		oveq1 7433	. . . . . . . 8 ⊢ (𝑁 = 1 → (𝑁 − 1) = (1 − 1))
11	10	oveq1d 7441	. . . . . . 7 ⊢ (𝑁 = 1 → ((𝑁 − 1) / 2) = ((1 − 1) / 2))
12	11	eleq1d 2814	. . . . . 6 ⊢ (𝑁 = 1 → (((𝑁 − 1) / 2) ∈ ℕ₀ ↔ ((1 − 1) / 2) ∈ ℕ₀))
13	12	adantr 479	. . . . 5 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀)) → (((𝑁 − 1) / 2) ∈ ℕ₀ ↔ ((1 − 1) / 2) ∈ ℕ₀))
14	9, 13	mpbiri 257	. . . 4 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀)) → ((𝑁 − 1) / 2) ∈ ℕ₀)
15	14	ex 411	. . 3 ⊢ (𝑁 = 1 → ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ₀))
16		2z 12632	. . . . . . . 8 ⊢ 2 ∈ ℤ
17	16	a1i 11	. . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀)) → 2 ∈ ℤ)
18		nn0z 12621	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
19	18	ad2antrl 726	. . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀)) → 𝑁 ∈ ℤ)
20		2re 12324	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
21		nn0re 12519	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
22		ltle 11340	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (2 < 𝑁 → 2 ≤ 𝑁))
23	20, 21, 22	sylancr 585	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (2 < 𝑁 → 2 ≤ 𝑁))
24	23	adantr 479	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → (2 < 𝑁 → 2 ≤ 𝑁))
25	24	impcom 406	. . . . . . 7 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀)) → 2 ≤ 𝑁)
26		eluz2 12866	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 2 ≤ 𝑁))
27	17, 19, 25, 26	syl3anbrc 1340	. . . . . 6 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀)) → 𝑁 ∈ (ℤ_≥‘2))
28		simprr 771	. . . . . 6 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀)) → ((𝑁 + 1) / 2) ∈ ℕ₀)
29	27, 28	jca 510	. . . . 5 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀)) → (𝑁 ∈ (ℤ_≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ₀))
30		nno 16366	. . . . 5 ⊢ ((𝑁 ∈ (ℤ_≥‘2) ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ)
31		nnnn0 12517	. . . . 5 ⊢ (((𝑁 − 1) / 2) ∈ ℕ → ((𝑁 − 1) / 2) ∈ ℕ₀)
32	29, 30, 31	3syl 18	. . . 4 ⊢ ((2 < 𝑁 ∧ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀)) → ((𝑁 − 1) / 2) ∈ ℕ₀)
33	32	ex 411	. . 3 ⊢ (2 < 𝑁 → ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ₀))
34	15, 33	jaoi 855	. 2 ⊢ ((𝑁 = 1 ∨ 2 < 𝑁) → ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ₀))
35	1, 34	mpcom 38	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) / 2) ∈ ℕ₀) → ((𝑁 − 1) / 2) ∈ ℕ₀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 < clt 11286 ≤ cle 11287 − cmin 11482 / cdiv 11909 ℕcn 12250 2c2 12305 ℕ₀cn0 12510 ℤcz 12596 ℤ_≥cuz 12860

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015

This theorem is referenced by: nn0ob 16368 nn0onn0ex 47674 nneom 47678 flnn0div2ge 47684 flnn0ohalf 47685

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