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Theorem prmirred 21789

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 21768	. . 3 ⊢ ℤ = (Base‘ℤ_ring)
3	1, 2	irredcl 20663	. 2 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ∈ ℤ)
4		elnn0 12615	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
5		zringring 21764	. . . . . . . . . . 11 ⊢ ℤ_ring ∈ Ring
6		zring0 21773	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘ℤ_ring)
7	1, 6	irredn0 20662	. . . . . . . . . . 11 ⊢ ((ℤ_ring ∈ Ring ∧ 𝐴 ∈ 𝐼) → 𝐴 ≠ 0)
8	5, 7	mpan 703	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ≠ 0)
9	8	necon2bi 2985	. . . . . . . . 9 ⊢ (𝐴 = 0 → ¬ 𝐴 ∈ 𝐼)
10	9	pm2.21d 122	. . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ))
11	10	jao1i 872	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ))
12	4, 11	sylbi 220	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ))
13		prmnn 16858	. . . . . . 7 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ ℙ → 𝐴 ∈ ℕ))
15	1	prmirredlem 21787	. . . . . . 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 382	. . . . 5 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ))
18		nn0re 12622	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℝ)
19		nn0ge0 12638	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 0 ≤ 𝐴)
20	18, 19	absidd 15599	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (abs‘𝐴) = 𝐴)
21	20	eleq1d 2845	. . . . 5 ⊢ (𝐴 ∈ ℕ₀ → ((abs‘𝐴) ∈ ℙ ↔ 𝐴 ∈ ℙ))
22	17, 21	bitr4d 285	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
23	22	adantl 487	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
24	1	prmirredlem 21787	. . . . . 6 ⊢ ( -𝐴 ∈ ℕ → ( -𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ))
25	24	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → ( -𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ))
26		eqid 2760	. . . . . . . . 9 ⊢ (inv_g‘ℤ_ring) = (inv_g‘ℤ_ring)
27	1, 26, 2	irrednegb 20670	. . . . . . . 8 ⊢ ((ℤ_ring ∈ Ring ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ 𝐼 ↔ ((inv_g‘ℤ_ring)‘𝐴) ∈ 𝐼))
28	5, 27	mpan 703	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ ((inv_g‘ℤ_ring)‘𝐴) ∈ 𝐼))
29		zsubrg 21735	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
30		subrgsubg 20838	. . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
31	29, 30	ax-mp 5	. . . . . . . . . 10 ⊢ ℤ ∈ (SubGrp‘ℂ_fld)
32		df-zring 21762	. . . . . . . . . . 11 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
33		eqid 2760	. . . . . . . . . . 11 ⊢ (inv_g‘ℂ_fld) = (inv_g‘ℂ_fld)
34	32, 33, 26	subginv 19352	. . . . . . . . . 10 ⊢ ((ℤ ∈ (SubGrp‘ℂ_fld) ∧ 𝐴 ∈ ℤ) → ((inv_g‘ℂ_fld)‘𝐴) = ((inv_g‘ℤ_ring)‘𝐴))
35	31, 34	mpan 703	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → ((inv_g‘ℂ_fld)‘𝐴) = ((inv_g‘ℤ_ring)‘𝐴))
36		zcn 12705	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
37		cnfldneg 21713	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((inv_g‘ℂ_fld)‘𝐴) = -𝐴)
38	36, 37	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → ((inv_g‘ℂ_fld)‘𝐴) = -𝐴)
39	35, 38	eqtr3d 2797	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → ((inv_g‘ℤ_ring)‘𝐴) = -𝐴)
40	39	eleq1d 2845	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (((inv_g‘ℤ_ring)‘𝐴) ∈ 𝐼 ↔ -𝐴 ∈ 𝐼))
41	28, 40	bitrd 282	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ -𝐴 ∈ 𝐼))
42	41	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (𝐴 ∈ 𝐼 ↔ -𝐴 ∈ 𝐼))
43		zre 12704	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
44	43	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 𝐴 ∈ ℝ)
45		nnnn0 12620	. . . . . . . . . 10 ⊢ ( -𝐴 ∈ ℕ → -𝐴 ∈ ℕ₀)
46	45	nn0ge0d 12677	. . . . . . . . 9 ⊢ ( -𝐴 ∈ ℕ → 0 ≤ -𝐴)
47	46	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 0 ≤ -𝐴)
48	44	le0neg1d 11894	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
49	47, 48	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 𝐴 ≤ 0)
50	44, 49	absnidd 15590	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (abs‘𝐴) = -𝐴)
51	50	eleq1d 2845	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → ((abs‘𝐴) ∈ ℙ ↔ -𝐴 ∈ ℙ))
52	25, 42, 51	3bitr4d 314	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
53	52	adantrl 729	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ)) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
54		elznn0nn 12714	. . . 4 ⊢ (𝐴 ∈ ℤ ↔ (𝐴 ∈ ℕ₀ ∨ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ)))
55	54	biimpi 219	. . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ ℕ₀ ∨ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ)))
56	23, 53, 55	mpjaodan 973	. 2 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
57	3, 56	biadanii 834	1 ⊢ (𝐴 ∈ 𝐼 ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) ∈ ℙ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ‘cfv 6543 ℂcc 11205 ℝcr 11206 0cc0 11207 ≤ cle 11351 -cneg 11549 ℕcn 12342 ℕ₀cn0 12613 ℤcz 12700 abscabs 15410 ℙcprime 16855 inv_gcminusg 19154 SubGrpcsubg 19339 Ringcrg 20468 Irredcir 20595 SubRingcsubrg 20830 ℂ_fldccnfld 21687 ℤ_ringczring 21761

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286 ax-mulf 11287

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-tpos 8251 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-er 8725 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-sup 9441 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-rp 13128 df-fz 13647 df-seq 14152 df-exp 14212 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-dvds 16432 df-prm 16856 df-gz 17117 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-0g 17621 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-grp 19156 df-minusg 19157 df-subg 19342 df-cmn 20005 df-abl 20006 df-mgp 20370 df-rng 20384 df-ur 20417 df-ring 20470 df-cring 20471 df-oppr 20576 df-dvdsr 20596 df-unit 20597 df-irred 20598 df-invr 20627 df-dvr 20640 df-subrng 20807 df-subrg 20831 df-drng 20991 df-cnfld 21688 df-zring 21762

This theorem is used by: (None)

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