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Mirrors > Home > MPE Home > Th. List > prmirred | Structured version Visualization version Unicode version |
Description: The irreducible elements of 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 | Irredℤring |
Ref | Expression |
---|---|
prmirred |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmirred.i | . . 3 Irredℤring | |
2 | zringbas 19824 | . . 3 ℤring | |
3 | 1, 2 | irredcl 18704 | . 2 |
4 | elnn0 11294 | . . . . . . 7 | |
5 | ax-1 6 | . . . . . . . 8 | |
6 | zringring 19821 | . . . . . . . . . . 11 ℤring | |
7 | zring0 19828 | . . . . . . . . . . . 12 ℤring | |
8 | 1, 7 | irredn0 18703 | . . . . . . . . . . 11 ℤring |
9 | 6, 8 | mpan 706 | . . . . . . . . . 10 |
10 | 9 | necon2bi 2824 | . . . . . . . . 9 |
11 | 10 | pm2.21d 118 | . . . . . . . 8 |
12 | 5, 11 | jaoi 394 | . . . . . . 7 |
13 | 4, 12 | sylbi 207 | . . . . . 6 |
14 | prmnn 15388 | . . . . . . 7 | |
15 | 14 | a1i 11 | . . . . . 6 |
16 | 1 | prmirredlem 19841 | . . . . . . 7 |
17 | 16 | a1i 11 | . . . . . 6 |
18 | 13, 15, 17 | pm5.21ndd 369 | . . . . 5 |
19 | nn0re 11301 | . . . . . . 7 | |
20 | nn0ge0 11318 | . . . . . . 7 | |
21 | 19, 20 | absidd 14161 | . . . . . 6 |
22 | 21 | eleq1d 2686 | . . . . 5 |
23 | 18, 22 | bitr4d 271 | . . . 4 |
24 | 23 | adantl 482 | . . 3 |
25 | 1 | prmirredlem 19841 | . . . . . 6 |
26 | 25 | adantl 482 | . . . . 5 |
27 | eqid 2622 | . . . . . . . . 9 ℤring ℤring | |
28 | 1, 27, 2 | irrednegb 18711 | . . . . . . . 8 ℤring ℤring |
29 | 6, 28 | mpan 706 | . . . . . . 7 ℤring |
30 | zsubrg 19799 | . . . . . . . . . . 11 SubRingℂfld | |
31 | subrgsubg 18786 | . . . . . . . . . . 11 SubRingℂfld SubGrpℂfld | |
32 | 30, 31 | ax-mp 5 | . . . . . . . . . 10 SubGrpℂfld |
33 | df-zring 19819 | . . . . . . . . . . 11 ℤring ℂfld ↾s | |
34 | eqid 2622 | . . . . . . . . . . 11 ℂfld ℂfld | |
35 | 33, 34, 27 | subginv 17601 | . . . . . . . . . 10 SubGrpℂfld ℂfld ℤring |
36 | 32, 35 | mpan 706 | . . . . . . . . 9 ℂfld ℤring |
37 | zcn 11382 | . . . . . . . . . 10 | |
38 | cnfldneg 19772 | . . . . . . . . . 10 ℂfld | |
39 | 37, 38 | syl 17 | . . . . . . . . 9 ℂfld |
40 | 36, 39 | eqtr3d 2658 | . . . . . . . 8 ℤring |
41 | 40 | eleq1d 2686 | . . . . . . 7 ℤring |
42 | 29, 41 | bitrd 268 | . . . . . 6 |
43 | 42 | adantr 481 | . . . . 5 |
44 | zre 11381 | . . . . . . . 8 | |
45 | 44 | adantr 481 | . . . . . . 7 |
46 | nnnn0 11299 | . . . . . . . . . 10 | |
47 | 46 | nn0ge0d 11354 | . . . . . . . . 9 |
48 | 47 | adantl 482 | . . . . . . . 8 |
49 | 45 | le0neg1d 10599 | . . . . . . . 8 |
50 | 48, 49 | mpbird 247 | . . . . . . 7 |
51 | 45, 50 | absnidd 14152 | . . . . . 6 |
52 | 51 | eleq1d 2686 | . . . . 5 |
53 | 26, 43, 52 | 3bitr4d 300 | . . . 4 |
54 | 53 | adantrl 752 | . . 3 |
55 | elznn0nn 11391 | . . . 4 | |
56 | 55 | biimpi 206 | . . 3 |
57 | 24, 54, 56 | mpjaodan 827 | . 2 |
58 | 3, 57 | biadan2 674 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wa 384 wceq 1483 wcel 1990 wne 2794 class class class wbr 4653 cfv 5888 cc 9934 cr 9935 cc0 9936 cle 10075 cneg 10267 cn 11020 cn0 11292 cz 11377 cabs 13974 cprime 15385 cminusg 17423 SubGrpcsubg 17588 crg 18547 Irredcir 18640 SubRingcsubrg 18776 ℂfldccnfld 19746 ℤringzring 19818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-prm 15386 df-gz 15634 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-irred 18643 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-cnfld 19747 df-zring 19819 |
This theorem is referenced by: (None) |
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