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Theorem prmirred 21381

Description: The irreducible elements of ℤ are exactly the prime numbers (and their negatives). (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by AV, 10-Jun-2019.)

**Hypothesis**
Ref	Expression
prmirred.i	⊢ 𝐼 = (Irred‘ℤ_ring)

**Assertion**
Ref	Expression
prmirred	⊢ (𝐴 ∈ 𝐼 ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) ∈ ℙ))

**Proof of Theorem prmirred**
Step	Hyp	Ref	Expression
1		prmirred.i	. . 3 ⊢ 𝐼 = (Irred‘ℤ_ring)
2		zringbas 21360	. . 3 ⊢ ℤ = (Base‘ℤ_ring)
3	1, 2	irredcl 20309	. 2 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ∈ ℤ)
4		elnn0 12386	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
5		zringring 21356	. . . . . . . . . . 11 ⊢ ℤ_ring ∈ Ring
6		zring0 21365	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘ℤ_ring)
7	1, 6	irredn0 20308	. . . . . . . . . . 11 ⊢ ((ℤ_ring ∈ Ring ∧ 𝐴 ∈ 𝐼) → 𝐴 ≠ 0)
8	5, 7	mpan 690	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ≠ 0)
9	8	necon2bi 2955	. . . . . . . . 9 ⊢ (𝐴 = 0 → ¬ 𝐴 ∈ 𝐼)
10	9	pm2.21d 121	. . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ))
11	10	jao1i 858	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ))
12	4, 11	sylbi 217	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ))
13		prmnn 16585	. . . . . . 7 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ ℙ → 𝐴 ∈ ℕ))
15	1	prmirredlem 21379	. . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ))
16	15	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ ℕ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ)))
17	12, 14, 16	pm5.21ndd 379	. . . . 5 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ))
18		nn0re 12393	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℝ)
19		nn0ge0 12409	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 0 ≤ 𝐴)
20	18, 19	absidd 15330	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (abs‘𝐴) = 𝐴)
21	20	eleq1d 2813	. . . . 5 ⊢ (𝐴 ∈ ℕ₀ → ((abs‘𝐴) ∈ ℙ ↔ 𝐴 ∈ ℙ))
22	17, 21	bitr4d 282	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
23	22	adantl 481	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
24	1	prmirredlem 21379	. . . . . 6 ⊢ (-𝐴 ∈ ℕ → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ))
25	24	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ))
26		eqid 2729	. . . . . . . . 9 ⊢ (inv_g‘ℤ_ring) = (inv_g‘ℤ_ring)
27	1, 26, 2	irrednegb 20316	. . . . . . . 8 ⊢ ((ℤ_ring ∈ Ring ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ 𝐼 ↔ ((inv_g‘ℤ_ring)‘𝐴) ∈ 𝐼))
28	5, 27	mpan 690	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ ((inv_g‘ℤ_ring)‘𝐴) ∈ 𝐼))
29		zsubrg 21327	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
30		subrgsubg 20462	. . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
31	29, 30	ax-mp 5	. . . . . . . . . 10 ⊢ ℤ ∈ (SubGrp‘ℂ_fld)
32		df-zring 21354	. . . . . . . . . . 11 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
33		eqid 2729	. . . . . . . . . . 11 ⊢ (inv_g‘ℂ_fld) = (inv_g‘ℂ_fld)
34	32, 33, 26	subginv 19012	. . . . . . . . . 10 ⊢ ((ℤ ∈ (SubGrp‘ℂ_fld) ∧ 𝐴 ∈ ℤ) → ((inv_g‘ℂ_fld)‘𝐴) = ((inv_g‘ℤ_ring)‘𝐴))
35	31, 34	mpan 690	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → ((inv_g‘ℂ_fld)‘𝐴) = ((inv_g‘ℤ_ring)‘𝐴))
36		zcn 12476	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
37		cnfldneg 21302	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((inv_g‘ℂ_fld)‘𝐴) = -𝐴)
38	36, 37	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → ((inv_g‘ℂ_fld)‘𝐴) = -𝐴)
39	35, 38	eqtr3d 2766	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → ((inv_g‘ℤ_ring)‘𝐴) = -𝐴)
40	39	eleq1d 2813	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (((inv_g‘ℤ_ring)‘𝐴) ∈ 𝐼 ↔ -𝐴 ∈ 𝐼))
41	28, 40	bitrd 279	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ -𝐴 ∈ 𝐼))
42	41	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (𝐴 ∈ 𝐼 ↔ -𝐴 ∈ 𝐼))
43		zre 12475	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
44	43	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 𝐴 ∈ ℝ)
45		nnnn0 12391	. . . . . . . . . 10 ⊢ (-𝐴 ∈ ℕ → -𝐴 ∈ ℕ₀)
46	45	nn0ge0d 12448	. . . . . . . . 9 ⊢ (-𝐴 ∈ ℕ → 0 ≤ -𝐴)
47	46	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 0 ≤ -𝐴)
48	44	le0neg1d 11691	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
49	47, 48	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 𝐴 ≤ 0)
50	44, 49	absnidd 15321	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (abs‘𝐴) = -𝐴)
51	50	eleq1d 2813	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → ((abs‘𝐴) ∈ ℙ ↔ -𝐴 ∈ ℙ))
52	25, 42, 51	3bitr4d 311	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
53	52	adantrl 716	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ)) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
54		elznn0nn 12485	. . . 4 ⊢ (𝐴 ∈ ℤ ↔ (𝐴 ∈ ℕ₀ ∨ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ)))
55	54	biimpi 216	. . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ ℕ₀ ∨ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ)))
56	23, 53, 55	mpjaodan 960	. 2 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
57	3, 56	biadanii 821	1 ⊢ (𝐴 ∈ 𝐼 ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) ∈ ℙ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5092 ‘cfv 6482 ℂcc 11007 ℝcr 11008 0cc0 11009 ≤ cle 11150 -cneg 11348 ℕcn 12128 ℕ₀cn0 12384 ℤcz 12471 abscabs 15141 ℙcprime 16582 inv_gcminusg 18813 SubGrpcsubg 18999 Ringcrg 20118 Irredcir 20241 SubRingcsubrg 20454 ℂ_fldccnfld 21261 ℤ_ringczring 21353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-rp 12894 df-fz 13411 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-prm 16583 df-gz 16842 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-0g 17345 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-grp 18815 df-minusg 18816 df-subg 19002 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-irred 20244 df-invr 20273 df-dvr 20286 df-subrng 20431 df-subrg 20455 df-drng 20616 df-cnfld 21262 df-zring 21354

This theorem is referenced by: (None)

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