elznn0 - Metamath Proof Explorer

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Theorem elznn0 11576

Description: Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004.)

**Assertion**
Ref	Expression
elznn0	⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ₀ ∨ -𝑁 ∈ ℕ₀)))

**Proof of Theorem elznn0**
Step	Hyp	Ref	Expression
1		elz 11563	. 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
2		elnn0 11478	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℝ → (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)))
4		elnn0 11478	. . . . . 6 ⊢ (-𝑁 ∈ ℕ₀ ↔ (-𝑁 ∈ ℕ ∨ -𝑁 = 0))
5		recn 10210	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ)
6		0cn 10216	. . . . . . . . 9 ⊢ 0 ∈ ℂ
7		negcon1 10517	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 0 ∈ ℂ) → (-𝑁 = 0 ↔ -0 = 𝑁))
8	5, 6, 7	sylancl 697	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (-𝑁 = 0 ↔ -0 = 𝑁))
9		neg0 10511	. . . . . . . . . 10 ⊢ -0 = 0
10	9	eqeq1i 2757	. . . . . . . . 9 ⊢ (-0 = 𝑁 ↔ 0 = 𝑁)
11		eqcom 2759	. . . . . . . . 9 ⊢ (0 = 𝑁 ↔ 𝑁 = 0)
12	10, 11	bitri 264	. . . . . . . 8 ⊢ (-0 = 𝑁 ↔ 𝑁 = 0)
13	8, 12	syl6bb 276	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (-𝑁 = 0 ↔ 𝑁 = 0))
14	13	orbi2d 740	. . . . . 6 ⊢ (𝑁 ∈ ℝ → ((-𝑁 ∈ ℕ ∨ -𝑁 = 0) ↔ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))
15	4, 14	syl5bb 272	. . . . 5 ⊢ (𝑁 ∈ ℝ → (-𝑁 ∈ ℕ₀ ↔ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))
16	3, 15	orbi12d 748	. . . 4 ⊢ (𝑁 ∈ ℝ → ((𝑁 ∈ ℕ₀ ∨ -𝑁 ∈ ℕ₀) ↔ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))))
17		3orass 1075	. . . . 5 ⊢ ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ (𝑁 = 0 ∨ (𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
18		orcom 401	. . . . 5 ⊢ ((𝑁 = 0 ∨ (𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ ((𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ∨ 𝑁 = 0))
19		orordir 554	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ∨ 𝑁 = 0) ↔ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))
20	17, 18, 19	3bitrri 287	. . . 4 ⊢ (((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))
21	16, 20	syl6rbb 277	. . 3 ⊢ (𝑁 ∈ ℝ → ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ (𝑁 ∈ ℕ₀ ∨ -𝑁 ∈ ℕ₀)))
22	21	pm5.32i 672	. 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ₀ ∨ -𝑁 ∈ ℕ₀)))
23	1, 22	bitri 264	1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ₀ ∨ -𝑁 ∈ ℕ₀)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∨ wo 382 ∧ wa 383 ∨ w3o 1071 = wceq 1624 ∈ wcel 2131 ℂcc 10118 ℝcr 10119 0cc0 10120 -cneg 10451 ℕcn 11204 ℕ₀cn0 11476 ℤcz 11561

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-po 5179 df-so 5180 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-ltxr 10263 df-sub 10452 df-neg 10453 df-n0 11477 df-z 11562

This theorem is referenced by: elz2 11578 zmulcl 11610 expnegz 13080 expaddzlem 13089 odd2np1 15259 mulgz 17761 mulgdirlem 17765 mulgdir 17766 mulgass 17772 mulgdi 18424 cxpmul2z 24628 rexzrexnn0 37862 pell1234qrdich 37919 pell14qrexpcl 37925 pell14qrdich 37927 rmxnn 38012 jm2.19lem4 38053

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