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Mirrors > Home > MPE Home > Th. List > prmirred | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
prmirred.i | ⊢ 𝐼 = (Irred‘ℤring) |
Ref | Expression |
---|---|
prmirred | ⊢ (𝐴 ∈ 𝐼 ↔ (𝐴 ∈ ℤ ∧ (abs‘𝐴) ∈ ℙ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmirred.i | . . 3 ⊢ 𝐼 = (Irred‘ℤring) | |
2 | zringbas 20828 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
3 | 1, 2 | irredcl 20086 | . 2 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ∈ ℤ) |
4 | elnn0 12374 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | |
5 | zringring 20825 | . . . . . . . . . . 11 ⊢ ℤring ∈ Ring | |
6 | zring0 20832 | . . . . . . . . . . . 12 ⊢ 0 = (0g‘ℤring) | |
7 | 1, 6 | irredn0 20085 | . . . . . . . . . . 11 ⊢ ((ℤring ∈ Ring ∧ 𝐴 ∈ 𝐼) → 𝐴 ≠ 0) |
8 | 5, 7 | mpan 689 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ≠ 0) |
9 | 8 | necon2bi 2973 | . . . . . . . . 9 ⊢ (𝐴 = 0 → ¬ 𝐴 ∈ 𝐼) |
10 | 9 | pm2.21d 121 | . . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ)) |
11 | 10 | jao1i 857 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ)) |
12 | 4, 11 | sylbi 216 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ)) |
13 | prmnn 16510 | . . . . . . 7 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ) | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)) |
15 | 1 | prmirredlem 20846 | . . . . . . 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 381 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ 𝐼 ↔ 𝐴 ∈ ℙ)) |
18 | nn0re 12381 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
19 | nn0ge0 12397 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴) | |
20 | 18, 19 | absidd 15267 | . . . . . 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 483 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ∈ ℕ0) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ)) |
24 | 1 | prmirredlem 20846 | . . . . . 6 ⊢ (-𝐴 ∈ ℕ → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ)) |
25 | 24 | adantl 483 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ)) |
26 | eqid 2738 | . . . . . . . . 9 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
27 | 1, 26, 2 | irrednegb 20093 | . . . . . . . 8 ⊢ ((ℤring ∈ Ring ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ 𝐼 ↔ ((invg‘ℤring)‘𝐴) ∈ 𝐼)) |
28 | 5, 27 | mpan 689 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ ((invg‘ℤring)‘𝐴) ∈ 𝐼)) |
29 | zsubrg 20803 | . . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
30 | subrgsubg 20181 | . . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubGrp‘ℂfld)) | |
31 | 29, 30 | ax-mp 5 | . . . . . . . . . 10 ⊢ ℤ ∈ (SubGrp‘ℂfld) |
32 | df-zring 20823 | . . . . . . . . . . 11 ⊢ ℤring = (ℂfld ↾s ℤ) | |
33 | eqid 2738 | . . . . . . . . . . 11 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
34 | 32, 33, 26 | subginv 18894 | . . . . . . . . . 10 ⊢ ((ℤ ∈ (SubGrp‘ℂfld) ∧ 𝐴 ∈ ℤ) → ((invg‘ℂfld)‘𝐴) = ((invg‘ℤring)‘𝐴)) |
35 | 31, 34 | mpan 689 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → ((invg‘ℂfld)‘𝐴) = ((invg‘ℤring)‘𝐴)) |
36 | zcn 12463 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
37 | cnfldneg 20776 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((invg‘ℂfld)‘𝐴) = -𝐴) | |
38 | 36, 37 | syl 17 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → ((invg‘ℂfld)‘𝐴) = -𝐴) |
39 | 35, 38 | eqtr3d 2780 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → ((invg‘ℤring)‘𝐴) = -𝐴) |
40 | 39 | eleq1d 2823 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (((invg‘ℤring)‘𝐴) ∈ 𝐼 ↔ -𝐴 ∈ 𝐼)) |
41 | 28, 40 | bitrd 279 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ -𝐴 ∈ 𝐼)) |
42 | 41 | adantr 482 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (𝐴 ∈ 𝐼 ↔ -𝐴 ∈ 𝐼)) |
43 | zre 12462 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
44 | 43 | adantr 482 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 𝐴 ∈ ℝ) |
45 | nnnn0 12379 | . . . . . . . . . 10 ⊢ (-𝐴 ∈ ℕ → -𝐴 ∈ ℕ0) | |
46 | 45 | nn0ge0d 12435 | . . . . . . . . 9 ⊢ (-𝐴 ∈ ℕ → 0 ≤ -𝐴) |
47 | 46 | adantl 483 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 0 ≤ -𝐴) |
48 | 44 | le0neg1d 11685 | . . . . . . . 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 15258 | . . . . . 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 715 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ)) → (𝐴 ∈ 𝐼 ↔ (abs‘𝐴) ∈ ℙ)) |
54 | elznn0nn 12472 | . . . 4 ⊢ (𝐴 ∈ ℤ ↔ (𝐴 ∈ ℕ0 ∨ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ))) | |
55 | 54 | biimpi 215 | . . 3 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ ℕ0 ∨ (𝐴 ∈ ℝ ∧ -𝐴 ∈ ℕ))) |
56 | 23, 53, 55 | mpjaodan 958 | . 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 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 class class class wbr 5104 ‘cfv 6494 ℂcc 11008 ℝcr 11009 0cc0 11010 ≤ cle 11149 -cneg 11345 ℕcn 12112 ℕ0cn0 12372 ℤcz 12458 abscabs 15079 ℙcprime 16507 invgcminusg 18709 SubGrpcsubg 18881 Ringcrg 19918 Irredcir 20022 SubRingcsubrg 20171 ℂfldccnfld 20749 ℤringczring 20822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087 ax-pre-sup 11088 ax-addf 11089 ax-mulf 11090 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-tpos 8150 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8607 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-sup 9337 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-div 11772 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-dec 12578 df-uz 12723 df-rp 12871 df-fz 13380 df-seq 13862 df-exp 13923 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-dvds 16097 df-prm 16508 df-gz 16762 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-starv 17108 df-tset 17112 df-ple 17113 df-ds 17115 df-unif 17116 df-0g 17283 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-grp 18711 df-minusg 18712 df-subg 18884 df-cmn 19523 df-mgp 19856 df-ur 19873 df-ring 19920 df-cring 19921 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-irred 20025 df-invr 20054 df-dvr 20065 df-drng 20140 df-subrg 20173 df-cnfld 20750 df-zring 20823 |
This theorem is referenced by: (None) |
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