Theorem opprmul 20621

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 20620	. . 3 ⊢ ∙ = tpos ·
6	5	oveqi 7464	. 2 ⊢ (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌)
7		ovtpos 8296	. 2 ⊢ (𝑋tpos · 𝑌) = (𝑌 · 𝑋)
8	6, 7	eqtri 2786	1 ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)

This proof depends on syntax axioms:   = wceq 1570  ‘cfv 6566  (class class class)co 7451  tpos ctpos 8280  Basecbs 17431  ._rcmulr 17473  opp_rcoppr 20617

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-1cn 11297  ax-addcl 11299

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-tpos 8281  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-nn 12376  df-2 12445  df-3 12446  df-sets 17386  df-slot 17404  df-ndx 17416  df-mulr 17487  df-oppr 20618

This theorem is used by:  crngoppr  20622  opprrng  20626  opprrngb  20627  opprring  20628  opprringb  20629  oppr1  20631  mulgass3  20634  opprunit  20658  unitmulcl  20661  unitgrp  20664  unitpropd  20698  opprirred  20703  irredlmul  20709  rhmopp  20810  opprsubrng  20862  subrguss  20890  subrgunit  20893  opprsubrg  20896  opprdomnb  21019  isdomn4r  21021  isdrng4  21043  isdrng2  21048  isdrngrd  21074  isdrngrdOLD  21076  srngmul  21160  issrngd  21163  rngridlmcl  21547  isridlrng  21549  isridl  21596  2idlcpblrng  21616  psropprmul  22610  invrvald  23046  isunit2  33899  opprlidlabs  34113  opprqusmulr  34119  qsdrngi  34123  ldualsmul  40458  lcdsmul  42925