Theorem opprmul 20579

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 20578	. . 3 ⊢ ∙ = tpos ·
6	5	oveqi 7437	. 2 ⊢ (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌)
7		ovtpos 8265	. 2 ⊢ (𝑋tpos · 𝑌) = (𝑌 · 𝑋)
8	6, 7	eqtri 2783	1 ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)

This proof depends on syntax axioms:   = wceq 1569  ‘cfv 6543  (class class class)co 7424  tpos ctpos 8249  Basecbs 17396  ._rcmulr 17438  opp_rcoppr 20575

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-1cn 11265  ax-addcl 11267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-2nd 8006  df-tpos 8250  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-nn 12343  df-2 12412  df-3 12413  df-sets 17351  df-slot 17369  df-ndx 17381  df-mulr 17451  df-oppr 20576

This theorem is used by:  crngoppr  20580  opprrng  20584  opprrngb  20585  opprring  20586  opprringb  20587  oppr1  20589  mulgass3  20592  opprunit  20616  unitmulcl  20619  unitgrp  20622  unitpropd  20656  opprirred  20661  irredlmul  20667  rhmopp  20768  opprsubrng  20820  subrguss  20848  subrgunit  20851  opprsubrg  20854  opprdomnb  20977  isdomn4r  20979  isdrng4  21001  isdrng2  21006  isdrngrd  21032  isdrngrdOLD  21034  srngmul  21118  issrngd  21121  rngridlmcl  21505  isridlrng  21507  isridl  21554  2idlcpblrng  21574  psropprmul  22564  invrvald  23000  isunit2  33850  opprlidlabs  34059  opprqusmulr  34065  qsdrngi  34069  ldualsmul  40301  lcdsmul  42768