Theorem opprmulg 13804

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
opprmulg	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋 ∙ 𝑌) = (𝑌 · 𝑋))

Proof of Theorem opprmulg

Step	Hyp	Ref	Expression
1		opprval.1	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
2		opprval.2	. . . . 5 ⊢ · = (.r‘𝑅)
3		opprval.3	. . . . 5 ⊢ 𝑂 = (oppr‘𝑅)
4		opprmulfval.4	. . . . 5 ⊢ ∙ = (.r‘𝑂)
5	1, 2, 3, 4	opprmulfvalg 13803	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∙ = tpos · )
6	5	oveqd 5960	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌))
7	6	3ad2ant1 1020	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌))
8		ovtposg 6344	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋tpos · 𝑌) = (𝑌 · 𝑋))
9	8	3adant1 1017	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋tpos · 𝑌) = (𝑌 · 𝑋))
10	7, 9	eqtrd 2237	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋 ∙ 𝑌) = (𝑌 · 𝑋))

Syntax hints:   → wi 4   ∧ w3a 980   = wceq 1372   ∈ wcel 2175  ‘cfv 5270  (class class class)co 5943  tpos ctpos 6329  Basecbs 12803  ._rcmulr 12881  opp_rcoppr 13800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-tpos 6330  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12806  df-slot 12807  df-sets 12810  df-mulr 12894  df-oppr 13801

This theorem is referenced by:  crngoppr  13805  opprrng  13810  opprrngbg  13811  opprring  13812  opprringbg  13813  oppr1g  13815  mulgass3  13818  opprunitd  13843  unitmulcl  13846  unitgrp  13849  unitpropdg  13881  rhmopp  13909  opprsubrngg  13944  subrguss  13969  subrgunit  13972  opprdomnbg  14007  isridlrng  14215  isridl  14237  2idlcpblrng  14256