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Theorem prmirred 21433

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 21412	. . 3 ⊢ ℤ = (Base‘ℤ_ring)
3	1, 2	irredcl 20382	. 2 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ∈ ℤ)
4		elnn0 12501	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
5		zringring 21408	. . . . . . . . . . 11 ⊢ ℤ_ring ∈ Ring
6		zring0 21417	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘ℤ_ring)
7	1, 6	irredn0 20381	. . . . . . . . . . 11 ⊢ ((ℤ_ring ∈ Ring ∧ 𝐴 ∈ 𝐼) → 𝐴 ≠ 0)
8	5, 7	mpan 690	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ≠ 0)
9	8	necon2bi 2962	. . . . . . . . 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 16691	. . . . . . 7 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ ℙ → 𝐴 ∈ ℕ))
15	1	prmirredlem 21431	. . . . . . 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 12508	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℝ)
19		nn0ge0 12524	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 0 ≤ 𝐴)
20	18, 19	absidd 15439	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (abs‘𝐴) = 𝐴)
21	20	eleq1d 2819	. . . . 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 21431	. . . . . 6 ⊢ (-𝐴 ∈ ℕ → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ))
25	24	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ))
26		eqid 2735	. . . . . . . . 9 ⊢ (inv_g‘ℤ_ring) = (inv_g‘ℤ_ring)
27	1, 26, 2	irrednegb 20389	. . . . . . . 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 21386	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
30		subrgsubg 20535	. . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
31	29, 30	ax-mp 5	. . . . . . . . . 10 ⊢ ℤ ∈ (SubGrp‘ℂ_fld)
32		df-zring 21406	. . . . . . . . . . 11 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
33		eqid 2735	. . . . . . . . . . 11 ⊢ (inv_g‘ℂ_fld) = (inv_g‘ℂ_fld)
34	32, 33, 26	subginv 19114	. . . . . . . . . 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 12591	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
37		cnfldneg 21356	. . . . . . . . . 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 2772	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → ((inv_g‘ℤ_ring)‘𝐴) = -𝐴)
40	39	eleq1d 2819	. . . . . . 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 12590	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
44	43	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 𝐴 ∈ ℝ)
45		nnnn0 12506	. . . . . . . . . 10 ⊢ (-𝐴 ∈ ℕ → -𝐴 ∈ ℕ₀)
46	45	nn0ge0d 12563	. . . . . . . . 9 ⊢ (-𝐴 ∈ ℕ → 0 ≤ -𝐴)
47	46	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 0 ≤ -𝐴)
48	44	le0neg1d 11806	. . . . . . . 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 15430	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (abs‘𝐴) = -𝐴)
51	50	eleq1d 2819	. . . . 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 12600	. . . 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 2108 ≠ wne 2932 class class class wbr 5119 ‘cfv 6530 ℂcc 11125 ℝcr 11126 0cc0 11127 ≤ cle 11268 -cneg 11465 ℕcn 12238 ℕ₀cn0 12499 ℤcz 12586 abscabs 15251 ℙcprime 16688 inv_gcminusg 18915 SubGrpcsubg 19101 Ringcrg 20191 Irredcir 20314 SubRingcsubrg 20527 ℂ_fldccnfld 21313 ℤ_ringczring 21405

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9452 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-rp 13007 df-fz 13523 df-seq 14018 df-exp 14078 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-dvds 16271 df-prm 16689 df-gz 16948 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-0g 17453 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-grp 18917 df-minusg 18918 df-subg 19104 df-cmn 19761 df-abl 19762 df-mgp 20099 df-rng 20111 df-ur 20140 df-ring 20193 df-cring 20194 df-oppr 20295 df-dvdsr 20315 df-unit 20316 df-irred 20317 df-invr 20346 df-dvr 20359 df-subrng 20504 df-subrg 20528 df-drng 20689 df-cnfld 21314 df-zring 21406

This theorem is referenced by: (None)

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