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Theorem prmirred 21449

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 21428	. . 3 ⊢ ℤ = (Base‘ℤ_ring)
3	1, 2	irredcl 20395	. 2 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ∈ ℤ)
4		elnn0 12430	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
5		zringring 21424	. . . . . . . . . . 11 ⊢ ℤ_ring ∈ Ring
6		zring0 21433	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘ℤ_ring)
7	1, 6	irredn0 20394	. . . . . . . . . . 11 ⊢ ((ℤ_ring ∈ Ring ∧ 𝐴 ∈ 𝐼) → 𝐴 ≠ 0)
8	5, 7	mpan 696	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ≠ 0)
9	8	necon2bi 2964	. . . . . . . . 9 ⊢ (𝐴 = 0 → ¬ 𝐴 ∈ 𝐼)
10	9	pm2.21d 121	. . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ))
11	10	jao1i 864	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ))
12	4, 11	sylbi 218	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ))
13		prmnn 16634	. . . . . . 7 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)
14	13	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ ℙ → 𝐴 ∈ ℕ))
15	1	prmirredlem 21447	. . . . . . 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 380	. . . . 5 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ))
18		nn0re 12437	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℝ)
19		nn0ge0 12453	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 0 ≤ 𝐴)
20	18, 19	absidd 15376	. . . . . 6 ⊢ (𝐴 ∈ ℕ₀ → (abs‘𝐴) = 𝐴)
21	20	eleq1d 2824	. . . . 5 ⊢ (𝐴 ∈ ℕ₀ → ((abs‘𝐴) ∈ ℙ ↔ 𝐴 ∈ ℙ))
22	17, 21	bitr4d 283	. . . 4 ⊢ (𝐴 ∈ ℕ₀ → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
23	22	adantl 482	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
24	1	prmirredlem 21447	. . . . . 6 ⊢ (-𝐴 ∈ ℕ → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ))
25	24	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ))
26		eqid 2739	. . . . . . . . 9 ⊢ (inv_g‘ℤ_ring) = (inv_g‘ℤ_ring)
27	1, 26, 2	irrednegb 20402	. . . . . . . 8 ⊢ ((ℤ_ring ∈ Ring ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ 𝐼 ↔ ((inv_g‘ℤ_ring)‘𝐴) ∈ 𝐼))
28	5, 27	mpan 696	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ ((inv_g‘ℤ_ring)‘𝐴) ∈ 𝐼))
29		zsubrg 21395	. . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
30		subrgsubg 20549	. . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
31	29, 30	ax-mp 5	. . . . . . . . . 10 ⊢ ℤ ∈ (SubGrp‘ℂ_fld)
32		df-zring 21422	. . . . . . . . . . 11 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
33		eqid 2739	. . . . . . . . . . 11 ⊢ (inv_g‘ℂ_fld) = (inv_g‘ℂ_fld)
34	32, 33, 26	subginv 19100	. . . . . . . . . 10 ⊢ ((ℤ ∈ (SubGrp‘ℂ_fld) ∧ 𝐴 ∈ ℤ) → ((inv_g‘ℂ_fld)‘𝐴) = ((inv_g‘ℤ_ring)‘𝐴))
35	31, 34	mpan 696	. . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → ((inv_g‘ℂ_fld)‘𝐴) = ((inv_g‘ℤ_ring)‘𝐴))
36		zcn 12520	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ)
37		cnfldneg 21373	. . . . . . . . . 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 2824	. . . . . . 7 ⊢ (𝐴 ∈ ℤ → (((inv_g‘ℤ_ring)‘𝐴) ∈ 𝐼 ↔ -𝐴 ∈ 𝐼))
41	28, 40	bitrd 280	. . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ -𝐴 ∈ 𝐼))
42	41	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (𝐴 ∈ 𝐼 ↔ -𝐴 ∈ 𝐼))
43		zre 12519	. . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ)
44	43	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 𝐴 ∈ ℝ)
45		nnnn0 12435	. . . . . . . . . 10 ⊢ (-𝐴 ∈ ℕ → -𝐴 ∈ ℕ₀)
46	45	nn0ge0d 12492	. . . . . . . . 9 ⊢ (-𝐴 ∈ ℕ → 0 ≤ -𝐴)
47	46	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 0 ≤ -𝐴)
48	44	le0neg1d 11712	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
49	47, 48	mpbird 258	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 𝐴 ≤ 0)
50	44, 49	absnidd 15367	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (abs‘𝐴) = -𝐴)
51	50	eleq1d 2824	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → ((abs‘𝐴) ∈ ℙ ↔ -𝐴 ∈ ℙ))
52	25, 42, 51	3bitr4d 312	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
53	52	adantrl 722	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ)) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
54		elznn0nn 12529	. . . 4 ⊢ (𝐴 ∈ ℤ ↔ (𝐴 ∈ ℕ₀ ∨ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ)))
55	54	biimpi 217	. . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ ℕ₀ ∨ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ)))
56	23, 53, 55	mpjaodan 966	. 2 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ))
57	3, 56	biadanii 827	1 ⊢ (𝐴 ∈ 𝐼 ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) ∈ ℙ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 class class class wbr 5072 ‘cfv 6485 ℂcc 11027 ℝcr 11028 0cc0 11029 ≤ cle 11171 -cneg 11369 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 abscabs 15187 ℙcprime 16631 inv_gcminusg 18901 SubGrpcsubg 19087 Ringcrg 20205 Irredcir 20327 SubRingcsubrg 20541 ℂ_fldccnfld 21347 ℤ_ringczring 21421

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-rp 12934 df-fz 13453 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16213 df-prm 16632 df-gz 16892 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904 df-subg 19090 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-irred 20330 df-invr 20359 df-dvr 20372 df-subrng 20518 df-subrg 20542 df-drng 20703 df-cnfld 21348 df-zring 21422

This theorem is referenced by: (None)

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