Theorem opprmul 20481

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 20480	. . 3 ⊢ ∙ = tpos ·
6	5	oveqi 7426	. 2 ⊢ (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌)
7		ovtpos 8239	. 2 ⊢ (𝑋tpos · 𝑌) = (𝑌 · 𝑋)
8	6, 7	eqtri 2783	1 ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)

This proof depends on syntax axioms:   = wceq 1570  ‘cfv 6533  (class class class)co 7413  tpos ctpos 8223  Basecbs 17301  ._rcmulr 17343  opp_rcoppr 20477

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-1cn 11182  ax-addcl 11184

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-2nd 7987  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-nn 12258  df-2 12327  df-3 12328  df-sets 17256  df-slot 17274  df-ndx 17286  df-mulr 17356  df-oppr 20478

This theorem is used by:  crngoppr  20482  opprrng  20486  opprrngb  20487  opprring  20488  opprringb  20489  oppr1  20491  mulgass3  20494  opprunit  20518  unitmulcl  20521  unitgrp  20524  unitpropd  20558  opprirred  20563  irredlmul  20569  rhmopp  20669  opprsubrng  20721  subrguss  20749  subrgunit  20752  opprsubrg  20755  opprdomnb  20878  isdomn4r  20880  isdrng4  20902  isdrng2  20906  isdrngrd  20932  isdrngrdOLD  20934  srngmul  21018  issrngd  21021  rngridlmcl  21405  isridlrng  21407  isridl  21454  2idlcpblrng  21473  psropprmul  22462  invrvald  22898  isunit2  33679  opprlidlabs  33887  opprqusmulr  33893  qsdrngi  33897  ldualsmul  40008  lcdsmul  42475