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Mirrors > Home > MPE Home > Th. List > opprmul | Structured version Visualization version GIF version |
Description: Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
opprval.1 | โข ๐ต = (Baseโ๐ ) |
opprval.2 | โข ยท = (.rโ๐ ) |
opprval.3 | โข ๐ = (opprโ๐ ) |
opprmulfval.4 | โข โ = (.rโ๐) |
Ref | Expression |
---|---|
opprmul | โข (๐ โ ๐) = (๐ ยท ๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprval.1 | . . . 4 โข ๐ต = (Baseโ๐ ) | |
2 | opprval.2 | . . . 4 โข ยท = (.rโ๐ ) | |
3 | opprval.3 | . . . 4 โข ๐ = (opprโ๐ ) | |
4 | opprmulfval.4 | . . . 4 โข โ = (.rโ๐) | |
5 | 1, 2, 3, 4 | opprmulfval 20152 | . . 3 โข โ = tpos ยท |
6 | 5 | oveqi 7422 | . 2 โข (๐ โ ๐) = (๐tpos ยท ๐) |
7 | ovtpos 8226 | . 2 โข (๐tpos ยท ๐) = (๐ ยท ๐) | |
8 | 6, 7 | eqtri 2761 | 1 โข (๐ โ ๐) = (๐ ยท ๐) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โcfv 6544 (class class class)co 7409 tpos ctpos 8210 Basecbs 17144 .rcmulr 17198 opprcoppr 20149 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-1cn 11168 ax-addcl 11170 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-nn 12213 df-2 12275 df-3 12276 df-sets 17097 df-slot 17115 df-ndx 17127 df-mulr 17211 df-oppr 20150 |
This theorem is referenced by: crngoppr 20154 opprring 20161 opprringb 20162 oppr1 20164 mulgass3 20167 opprunit 20191 unitmulcl 20194 unitgrp 20197 unitpropd 20231 opprirred 20236 irredlmul 20242 rhmopp 20288 subrguss 20334 subrgunit 20337 opprsubrg 20340 isdrng2 20371 isdrngrd 20391 isdrngrdOLD 20393 srngmul 20466 issrngd 20469 isridl 20859 2idlcpbl 20871 opprdomn 20919 psropprmul 21760 invrvald 22178 isdrng4 32395 opprlidlabs 32599 opprqusmulr 32605 qsdrngi 32609 ldualsmul 38005 lcdsmul 40473 opprrng 46674 opprrngb 46675 opprsubrng 46738 rngridlmcl 46749 isridlrng 46751 2idlcpblrng 46766 |
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