elznn0 - Metamath Proof Explorer

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Theorem elznn0 11269

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 11256	. 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)))
2		elnn0 11171	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
3	2	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℝ → (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)))
4		elnn0 11171	. . . . . 6 ⊢ (-𝑁 ∈ ℕ₀ ↔ (-𝑁 ∈ ℕ ∨ -𝑁 = 0))
5		recn 9905	. . . . . . . . 9 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℂ)
6		0cn 9911	. . . . . . . . 9 ⊢ 0 ∈ ℂ
7		negcon1 10212	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 0 ∈ ℂ) → (-𝑁 = 0 ↔ -0 = 𝑁))
8	5, 6, 7	sylancl 693	. . . . . . . 8 ⊢ (𝑁 ∈ ℝ → (-𝑁 = 0 ↔ -0 = 𝑁))
9		neg0 10206	. . . . . . . . . 10 ⊢ -0 = 0
10	9	eqeq1i 2615	. . . . . . . . 9 ⊢ (-0 = 𝑁 ↔ 0 = 𝑁)
11		eqcom 2617	. . . . . . . . 9 ⊢ (0 = 𝑁 ↔ 𝑁 = 0)
12	10, 11	bitri 263	. . . . . . . 8 ⊢ (-0 = 𝑁 ↔ 𝑁 = 0)
13	8, 12	syl6bb 275	. . . . . . 7 ⊢ (𝑁 ∈ ℝ → (-𝑁 = 0 ↔ 𝑁 = 0))
14	13	orbi2d 734	. . . . . 6 ⊢ (𝑁 ∈ ℝ → ((-𝑁 ∈ ℕ ∨ -𝑁 = 0) ↔ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))
15	4, 14	syl5bb 271	. . . . 5 ⊢ (𝑁 ∈ ℝ → (-𝑁 ∈ ℕ₀ ↔ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))
16	3, 15	orbi12d 742	. . . 4 ⊢ (𝑁 ∈ ℝ → ((𝑁 ∈ ℕ₀ ∨ -𝑁 ∈ ℕ₀) ↔ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0))))
17		3orass 1034	. . . . 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 552	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ∨ 𝑁 = 0) ↔ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)))
20	17, 18, 19	3bitrri 286	. . . 4 ⊢ (((𝑁 ∈ ℕ ∨ 𝑁 = 0) ∨ (-𝑁 ∈ ℕ ∨ 𝑁 = 0)) ↔ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))
21	16, 20	syl6rbb 276	. . 3 ⊢ (𝑁 ∈ ℝ → ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) ↔ (𝑁 ∈ ℕ₀ ∨ -𝑁 ∈ ℕ₀)))
22	21	pm5.32i 667	. 2 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ₀ ∨ -𝑁 ∈ ℕ₀)))
23	1, 22	bitri 263	1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 ∈ ℕ₀ ∨ -𝑁 ∈ ℕ₀)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 195 ∨ wo 382 ∧ wa 383 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 ℂcc 9813 ℝcr 9814 0cc0 9815 -cneg 10146 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 df-n0 11170 df-z 11255

This theorem is referenced by: elz2 11271 zmulcl 11303 expnegz 12756 expaddzlem 12765 odd2np1 14903 mulgz 17391 mulgdirlem 17395 mulgdir 17396 mulgass 17402 mulgdi 18055 cxpmul2z 24237 rexzrexnn0 36386 pell1234qrdich 36443 pell14qrexpcl 36449 pell14qrdich 36451 rmxnn 36536 jm2.19lem4 36577

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