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Theorem opprmul 19376
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.)
Hypotheses
Ref Expression
opprval.1 𝐵 = (Base‘𝑅)
opprval.2 · = (.r𝑅)
opprval.3 𝑂 = (oppr𝑅)
opprmulfval.4 = (.r𝑂)
Assertion
Ref Expression
opprmul (𝑋 𝑌) = (𝑌 · 𝑋)

Proof of Theorem opprmul
StepHypRef Expression
1 opprval.1 . . . 4 𝐵 = (Base‘𝑅)
2 opprval.2 . . . 4 · = (.r𝑅)
3 opprval.3 . . . 4 𝑂 = (oppr𝑅)
4 opprmulfval.4 . . . 4 = (.r𝑂)
51, 2, 3, 4opprmulfval 19375 . . 3 = tpos ·
65oveqi 7152 . 2 (𝑋 𝑌) = (𝑋tpos · 𝑌)
7 ovtpos 7894 . 2 (𝑋tpos · 𝑌) = (𝑌 · 𝑋)
86, 7eqtri 2824 1 (𝑋 𝑌) = (𝑌 · 𝑋)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cfv 6328  (class class class)co 7139  tpos ctpos 7878  Basecbs 16479  .rcmulr 16562  opprcoppr 19372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-1cn 10588  ax-addcl 10590
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-nn 11630  df-2 11692  df-3 11693  df-ndx 16482  df-slot 16483  df-sets 16486  df-mulr 16575  df-oppr 19373
This theorem is referenced by:  crngoppr  19377  opprring  19381  opprringb  19382  oppr1  19384  mulgass3  19387  opprunit  19411  unitmulcl  19414  unitgrp  19417  unitpropd  19447  opprirred  19452  irredlmul  19458  isdrng2  19509  isdrngrd  19525  subrguss  19547  subrgunit  19550  opprsubrg  19553  srngmul  19626  issrngd  19629  2idlcpbl  20004  opprdomn  20071  psropprmul  20871  invrvald  21285  rhmopp  30947  ldualsmul  36430  lcdsmul  38897
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