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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 21464 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
| 3 | 1, 2 | irredcl 20424 | . 2 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ∈ ℤ) |
| 4 | elnn0 12528 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0)) | |
| 5 | zringring 21460 | . . . . . . . . . . 11 ⊢ ℤring ∈ Ring | |
| 6 | zring0 21469 | . . . . . . . . . . . 12 ⊢ 0 = (0g‘ℤring) | |
| 7 | 1, 6 | irredn0 20423 | . . . . . . . . . . 11 ⊢ ((ℤring ∈ Ring ∧ 𝐴 ∈ 𝐼) → 𝐴 ≠ 0) |
| 8 | 5, 7 | mpan 690 | . . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐼 → 𝐴 ≠ 0) |
| 9 | 8 | necon2bi 2971 | . . . . . . . . 9 ⊢ (𝐴 = 0 → ¬ 𝐴 ∈ 𝐼) |
| 10 | 9 | pm2.21d 121 | . . . . . . . 8 ⊢ (𝐴 = 0 → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ)) |
| 11 | 10 | jao1i 859 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ)) |
| 12 | 4, 11 | sylbi 217 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ 𝐼 → 𝐴 ∈ ℕ)) |
| 13 | prmnn 16711 | . . . . . . 7 ⊢ (𝐴 ∈ ℙ → 𝐴 ∈ ℕ) | |
| 14 | 13 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → (𝐴 ∈ ℙ → 𝐴 ∈ ℕ)) |
| 15 | 1 | prmirredlem 21483 | . . . . . . 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 12535 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
| 19 | nn0ge0 12551 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → 0 ≤ 𝐴) | |
| 20 | 18, 19 | absidd 15461 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → (abs‘𝐴) = 𝐴) |
| 21 | 20 | eleq1d 2826 | . . . . 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 21483 | . . . . . 6 ⊢ (-𝐴 ∈ ℕ → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ)) |
| 25 | 24 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (-𝐴 ∈ 𝐼 ↔ -𝐴 ∈ ℙ)) |
| 26 | eqid 2737 | . . . . . . . . 9 ⊢ (invg‘ℤring) = (invg‘ℤring) | |
| 27 | 1, 26, 2 | irrednegb 20431 | . . . . . . . 8 ⊢ ((ℤring ∈ Ring ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ 𝐼 ↔ ((invg‘ℤring)‘𝐴) ∈ 𝐼)) |
| 28 | 5, 27 | mpan 690 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (𝐴 ∈ 𝐼 ↔ ((invg‘ℤring)‘𝐴) ∈ 𝐼)) |
| 29 | zsubrg 21438 | . . . . . . . . . . 11 ⊢ ℤ ∈ (SubRing‘ℂfld) | |
| 30 | subrgsubg 20577 | . . . . . . . . . . 11 ⊢ (ℤ ∈ (SubRing‘ℂfld) → ℤ ∈ (SubGrp‘ℂfld)) | |
| 31 | 29, 30 | ax-mp 5 | . . . . . . . . . 10 ⊢ ℤ ∈ (SubGrp‘ℂfld) |
| 32 | df-zring 21458 | . . . . . . . . . . 11 ⊢ ℤring = (ℂfld ↾s ℤ) | |
| 33 | eqid 2737 | . . . . . . . . . . 11 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
| 34 | 32, 33, 26 | subginv 19151 | . . . . . . . . . 10 ⊢ ((ℤ ∈ (SubGrp‘ℂfld) ∧ 𝐴 ∈ ℤ) → ((invg‘ℂfld)‘𝐴) = ((invg‘ℤring)‘𝐴)) |
| 35 | 31, 34 | mpan 690 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → ((invg‘ℂfld)‘𝐴) = ((invg‘ℤring)‘𝐴)) |
| 36 | zcn 12618 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
| 37 | cnfldneg 21408 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((invg‘ℂfld)‘𝐴) = -𝐴) | |
| 38 | 36, 37 | syl 17 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → ((invg‘ℂfld)‘𝐴) = -𝐴) |
| 39 | 35, 38 | eqtr3d 2779 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → ((invg‘ℤring)‘𝐴) = -𝐴) |
| 40 | 39 | eleq1d 2826 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → (((invg‘ℤring)‘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 12617 | . . . . . . . 8 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
| 44 | 43 | adantr 480 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 𝐴 ∈ ℝ) |
| 45 | nnnn0 12533 | . . . . . . . . . 10 ⊢ (-𝐴 ∈ ℕ → -𝐴 ∈ ℕ0) | |
| 46 | 45 | nn0ge0d 12590 | . . . . . . . . 9 ⊢ (-𝐴 ∈ ℕ → 0 ≤ -𝐴) |
| 47 | 46 | adantl 481 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → 0 ≤ -𝐴) |
| 48 | 44 | le0neg1d 11834 | . . . . . . . 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 15452 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ -𝐴 ∈ ℕ) → (abs‘𝐴) = -𝐴) |
| 51 | 50 | eleq1d 2826 | . . . . 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 12627 | . . . 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 961 | . 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 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 ℂcc 11153 ℝcr 11154 0cc0 11155 ≤ cle 11296 -cneg 11493 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 abscabs 15273 ℙcprime 16708 invgcminusg 18952 SubGrpcsubg 19138 Ringcrg 20230 Irredcir 20356 SubRingcsubrg 20569 ℂfldccnfld 21364 ℤringczring 21457 |
| 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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-prm 16709 df-gz 16968 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-irred 20359 df-invr 20388 df-dvr 20401 df-subrng 20546 df-subrg 20570 df-drng 20731 df-cnfld 21365 df-zring 21458 |
| This theorem is referenced by: (None) |
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