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Theorem prmirred 21502

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 21481	. . 3 ⊢ ℤ = (Base‘ℤ_ring)
3	1, 2	irredcl 20440	. 2 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ∈ ℤ)
4		elnn0 12525	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
5		zringring 21477	. . . . . . . . . . 11 ⊢ ℤ_ring ∈ Ring
6		zring0 21486	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘ℤ_ring)
7	1, 6	irredn0 20439	. . . . . . . . . . 11 ⊢ ((ℤ_ring ∈ Ring ∧ 𝐴 ∈ 𝐼) → 𝐴 ≠ 0)
8	5, 7	mpan 690	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ≠ 0)
9	8	necon2bi 2968	. . . . . . . . 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 16707	. . . . . . 7 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ ℙ → 𝐴 ∈ ℕ))
15	1	prmirredlem 21500	. . . . . . 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 12532	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℝ)
19		nn0ge0 12548	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 0 ≤ 𝐴)
20	18, 19	absidd 15457	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (abs‘𝐴) = 𝐴)
21	20	eleq1d 2823	. . . . 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 21500	. . . . . 6 ⊢ (-𝐴 ∈ ℕ → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ))
25	24	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ))
26		eqid 2734	. . . . . . . . 9 ⊢ (inv_g‘ℤ_ring) = (inv_g‘ℤ_ring)
27	1, 26, 2	irrednegb 20447	. . . . . . . 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 21455	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
30		subrgsubg 20593	. . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
31	29, 30	ax-mp 5	. . . . . . . . . 10 ⊢ ℤ ∈ (SubGrp‘ℂ_fld)
32		df-zring 21475	. . . . . . . . . . 11 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
33		eqid 2734	. . . . . . . . . . 11 ⊢ (inv_g‘ℂ_fld) = (inv_g‘ℂ_fld)
34	32, 33, 26	subginv 19163	. . . . . . . . . 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 12615	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
37		cnfldneg 21425	. . . . . . . . . 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 2776	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → ((inv_g‘ℤ_ring)‘𝐴) = -𝐴)
40	39	eleq1d 2823	. . . . . . 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 12614	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
44	43	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 𝐴 ∈ ℝ)
45		nnnn0 12530	. . . . . . . . . 10 ⊢ (-𝐴 ∈ ℕ → -𝐴 ∈ ℕ₀)
46	45	nn0ge0d 12587	. . . . . . . . 9 ⊢ (-𝐴 ∈ ℕ → 0 ≤ -𝐴)
47	46	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 0 ≤ -𝐴)
48	44	le0neg1d 11831	. . . . . . . 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 15448	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (abs‘𝐴) = -𝐴)
51	50	eleq1d 2823	. . . . 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 12624	. . . 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 822	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 1536 ∈ wcel 2105 ≠ wne 2937 class class class wbr 5147 ‘cfv 6562 ℂcc 11150 ℝcr 11151 0cc0 11152 ≤ cle 11293 -cneg 11490 ℕcn 12263 ℕ₀cn0 12523 ℤcz 12610 abscabs 15269 ℙcprime 16704 inv_gcminusg 18964 SubGrpcsubg 19150 Ringcrg 20250 Irredcir 20372 SubRingcsubrg 20585 ℂ_fldccnfld 21381 ℤ_ringczring 21474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-prm 16705 df-gz 16963 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-subg 19153 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-irred 20375 df-invr 20404 df-dvr 20417 df-subrng 20562 df-subrg 20586 df-drng 20747 df-cnfld 21382 df-zring 21475

This theorem is referenced by: (None)

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