Theorem opprmul 20249

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	⊢ · = (.r‘𝑅)
opprval.3	⊢ 𝑂 = (oppr‘𝑅)
opprmulfval.4	⊢ ∙ = (.r‘𝑂)

Assertion

Ref	Expression
opprmul	⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)

Proof of Theorem opprmul

Step	Hyp	Ref	Expression
1		opprval.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
2		opprval.2	. . . 4 ⊢ · = (.r‘𝑅)
3		opprval.3	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
4		opprmulfval.4	. . . 4 ⊢ ∙ = (.r‘𝑂)
5	1, 2, 3, 4	opprmulfval 20248	. . 3 ⊢ ∙ = tpos ·
6	5	oveqi 7400	. 2 ⊢ (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌)
7		ovtpos 8220	. 2 ⊢ (𝑋tpos · 𝑌) = (𝑌 · 𝑋)
8	6, 7	eqtri 2752	1 ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)

Syntax hints:   = wceq 1540  ‘cfv 6511  (class class class)co 7387  tpos ctpos 8204  Basecbs 17179  ._rcmulr 17221  opp_rcoppr 20245

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-1cn 11126  ax-addcl 11128

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-nn 12187  df-2 12249  df-3 12250  df-sets 17134  df-slot 17152  df-ndx 17164  df-mulr 17234  df-oppr 20246

This theorem is referenced by:  crngoppr  20250  opprrng  20254  opprrngb  20255  opprring  20256  opprringb  20257  oppr1  20259  mulgass3  20262  opprunit  20286  unitmulcl  20289  unitgrp  20292  unitpropd  20326  opprirred  20331  irredlmul  20337  rhmopp  20418  opprsubrng  20468  subrguss  20496  subrgunit  20499  opprsubrg  20502  opprdomnb  20626  isdomn4r  20628  isdrng2  20652  isdrngrd  20675  isdrngrdOLD  20677  srngmul  20761  issrngd  20764  rngridlmcl  21127  isridlrng  21129  isridl  21162  2idlcpblrng  21181  psropprmul  22122  invrvald  22563  isunit2  33191  isdrng4  33245  opprlidlabs  33456  opprqusmulr  33462  qsdrngi  33466  ldualsmul  39128  lcdsmul  41596