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Mirrors > Home > MPE Home > Th. List > opprirred | Structured version Visualization version Unicode version |
Description: Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
opprirred.1 | oppr |
opprirred.2 | Irred |
Ref | Expression |
---|---|
opprirred | Irred |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralcom 3098 | . . . . 5 Unit Unit Unit Unit | |
2 | eqid 2622 | . . . . . . . 8 | |
3 | eqid 2622 | . . . . . . . 8 | |
4 | opprirred.1 | . . . . . . . 8 oppr | |
5 | eqid 2622 | . . . . . . . 8 | |
6 | 2, 3, 4, 5 | opprmul 18626 | . . . . . . 7 |
7 | 6 | neeq1i 2858 | . . . . . 6 |
8 | 7 | 2ralbii 2981 | . . . . 5 Unit Unit Unit Unit |
9 | 1, 8 | bitr4i 267 | . . . 4 Unit Unit Unit Unit |
10 | 9 | anbi2i 730 | . . 3 Unit Unit Unit Unit Unit Unit |
11 | eqid 2622 | . . . 4 Unit Unit | |
12 | opprirred.2 | . . . 4 Irred | |
13 | eqid 2622 | . . . 4 Unit Unit | |
14 | 2, 11, 12, 13, 3 | isirred 18699 | . . 3 Unit Unit Unit |
15 | 4, 2 | opprbas 18629 | . . . 4 |
16 | 11, 4 | opprunit 18661 | . . . 4 Unit Unit |
17 | eqid 2622 | . . . 4 Irred Irred | |
18 | 15, 16, 17, 13, 5 | isirred 18699 | . . 3 Irred Unit Unit Unit |
19 | 10, 14, 18 | 3bitr4i 292 | . 2 Irred |
20 | 19 | eqriv 2619 | 1 Irred |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wceq 1483 wcel 1990 wne 2794 wral 2912 cdif 3571 cfv 5888 (class class class)co 6650 cbs 15857 cmulr 15942 opprcoppr 18622 Unitcui 18639 Irredcir 18640 |
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 |
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-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-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-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgp 18490 df-ur 18502 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-irred 18643 |
This theorem is referenced by: irredlmul 18708 |
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