Theorem opprmul 20418

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 20417	. . 3 ⊢ ∙ = tpos ·
6	5	oveqi 7423	. 2 ⊢ (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌)
7		ovtpos 8233	. 2 ⊢ (𝑋tpos · 𝑌) = (𝑌 · 𝑋)
8	6, 7	eqtri 2786	1 ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)

Syntax hints:   = wceq 1570  ‘cfv 6536  (class class class)co 7410  tpos ctpos 8217  Basecbs 17264  ._rcmulr 17306  opp_rcoppr 20414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-1cn 11153  ax-addcl 11155

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-nn 12229  df-2 12298  df-3 12299  df-sets 17219  df-slot 17237  df-ndx 17249  df-mulr 17319  df-oppr 20415

This theorem is referenced by:  crngoppr  20419  opprrng  20423  opprrngb  20424  opprring  20425  opprringb  20426  oppr1  20428  mulgass3  20431  opprunit  20455  unitmulcl  20458  unitgrp  20461  unitpropd  20495  opprirred  20500  irredlmul  20506  rhmopp  20606  opprsubrng  20658  subrguss  20686  subrgunit  20689  opprsubrg  20692  opprdomnb  20815  isdomn4r  20817  isdrng4  20839  isdrng2  20843  isdrngrd  20869  isdrngrdOLD  20871  srngmul  20955  issrngd  20958  rngridlmcl  21342  isridlrng  21344  isridl  21391  2idlcpblrng  21410  psropprmul  22397  invrvald  22833  isunit2  33559  opprlidlabs  33767  opprqusmulr  33773  qsdrngi  33777  ldualsmul  39909  lcdsmul  42376