elnnz - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > elnnz		Structured version Visualization version GIF version

Theorem elnnz 11985

Description: Positive integer property expressed in terms of integers. (Contributed by NM, 8-Jan-2002.)

**Assertion**
Ref	Expression
elnnz	⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))

**Proof of Theorem elnnz**
Step	Hyp	Ref	Expression
1		nnre 11639	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
2		orc 863	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))
3		nngt0 11662	. . . 4 ⊢ (𝑁 ∈ ℕ → 0 < 𝑁)
4	1, 2, 3	jca31 517	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))) ∧ 0 < 𝑁))
5		idd 24	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ 0 < 𝑁) → (𝑁 ∈ ℕ → 𝑁 ∈ ℕ))
6		lt0neg2 11141	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (0 < 𝑁 ↔ -𝑁 < 0))
7		renegcl 10943	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → -𝑁 ∈ ℝ)
8		0re 10637	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
9		ltnsym 10732	. . . . . . . . . . . . 13 ⊢ ((-𝑁 ∈ ℝ ∧ 0 ∈ ℝ) → (-𝑁 < 0 → ¬ 0 < -𝑁))
10	7, 8, 9	sylancl 588	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → (-𝑁 < 0 → ¬ 0 < -𝑁))
11	6, 10	sylbid 242	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (0 < 𝑁 → ¬ 0 < -𝑁))
12	11	imp 409	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℝ ∧ 0 < 𝑁) → ¬ 0 < -𝑁)
13		nngt0 11662	. . . . . . . . . 10 ⊢ (-𝑁 ∈ ℕ → 0 < -𝑁)
14	12, 13	nsyl 142	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 0 < 𝑁) → ¬ -𝑁 ∈ ℕ)
15		gt0ne0 11099	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℝ ∧ 0 < 𝑁) → 𝑁 ≠ 0)
16	15	neneqd 3021	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 0 < 𝑁) → ¬ 𝑁 = 0)
17		ioran 980	. . . . . . . . 9 ⊢ (¬ (-𝑁 ∈ ℕ ∨ 𝑁 = 0) ↔ (¬ -𝑁 ∈ ℕ ∧ ¬ 𝑁 = 0))
18	14, 16, 17	sylanbrc 585	. . . . . . . 8 ⊢ ((𝑁 ∈ ℝ ∧ 0 < 𝑁) → ¬ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))
19	18	pm2.21d 121	. . . . . . 7 ⊢ ((𝑁 ∈ ℝ ∧ 0 < 𝑁) → ((-𝑁 ∈ ℕ ∨ 𝑁 = 0) → 𝑁 ∈ ℕ))
20	5, 19	jaod 855	. . . . . 6 ⊢ ((𝑁 ∈ ℝ ∧ 0 < 𝑁) → ((𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)) → 𝑁 ∈ ℕ))
21	20	ex 415	. . . . 5 ⊢ (𝑁 ∈ ℝ → (0 < 𝑁 → ((𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)) → 𝑁 ∈ ℕ)))
22	21	com23 86	. . . 4 ⊢ (𝑁 ∈ ℝ → ((𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)) → (0 < 𝑁 → 𝑁 ∈ ℕ)))
23	22	imp31 420	. . 3 ⊢ (((𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))) ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
24	4, 23	impbii 211	. 2 ⊢ (𝑁 ∈ ℕ ↔ ((𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))) ∧ 0 < 𝑁))
25		elz 11977	. . . 4 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
26		3orrot 1088	. . . . . 6 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ (𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ ∨ 𝑁 = 0))
27		3orass 1086	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ ∨ 𝑁 = 0) ↔ (𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))
28	26, 27	bitri 277	. . . . 5 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ (𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))
29	28	anbi2i 624	. . . 4 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))))
30	25, 29	bitri 277	. . 3 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))))
31	30	anbi1i 625	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 0 < 𝑁) ↔ ((𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))) ∧ 0 < 𝑁))
32	24, 31	bitr4i 280	1 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∨ w3o 1082 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ℝcr 10530 0cc0 10531 < clt 10669 -cneg 10865 ℕcn 11632 ℤcz 11975

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-z 11976

This theorem is referenced by: elnn0z 11988 elnnz1 12002 znnsub 12022 nn0ge0div 12045 msqznn 12058 elpq 12368 lbfzo0 13071 elfzo0z 13073 fzofzim 13078 fzo1fzo0n0 13082 elfzodifsumelfzo 13097 elfznelfzo 13136 nnesq 13582 swrdlsw 14023 pfxccatin12lem3 14088 repswswrd 14140 cshwcsh2id 14184 swrd2lsw 14308 2swrd2eqwrdeq 14309 nnabscl 14679 iseralt 15035 sqrt2irrlem 15595 p1modz1 15608 nndivdvds 15610 oddge22np1 15692 evennn2n 15694 nno 15727 nnoddm1d2 15731 ndvdsadd 15755 bitsfzolem 15777 sqgcd 15903 qredeu 15996 prmind2 16023 qgt0numnn 16085 oddprm 16141 pythagtriplem6 16152 pythagtriplem11 16156 pythagtriplem13 16158 pythagtriplem19 16164 pc2dvds 16209 pcadd 16219 prmreclem3 16248 4sqlem11 16285 4sqlem12 16286 prmgaplem7 16387 cshwshashlem2 16424 subgmulg 18287 znidomb 20702 sgmnncl 25718 muinv 25764 mersenne 25797 bposlem6 25859 gausslemma2dlem1a 25935 lgseisenlem1 25945 lgsquadlem1 25950 lgsquadlem2 25951 2sqlem8 25996 2sqnn0 26008 dchrisum0flblem2 26079 clwlkclwwlklem2a2 27765 clwlkclwwlklem2a4 27769 clwlkclwwlklem2a 27770 eucrct2eupth1 28017 nn0prpwlem 33665 poimirlem7 34893 poimirlem29 34915 mblfinlem2 34924 rtprmirr 39187 dffltz 39264 irrapxlem4 39415 rmspecnonsq 39497 rmynn 39546 jm2.24 39553 jm2.23 39586 jm2.20nn 39587 jm2.27a 39595 jm2.27c 39597 rmydioph 39604 jm3.1lem3 39609 sumnnodd 41904 dvnxpaek 42220 dirkertrigeqlem3 42379 fourierdlem47 42432 fouriersw 42510 etransclem15 42528 etransclem24 42537 etransclem25 42538 etransclem35 42548 etransclem48 42561 zm1nn 43496 iccpartigtl 43577 nnoALTV 43854 nneven 43857 ztprmneprm 44389 blennngt2o2 44646

W3C validator