Theorem opprmulg 13567

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 13566	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∙ = tpos · )
6	5	oveqd 5935	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌))
7	6	3ad2ant1 1020	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌))
8		ovtposg 6312	. . 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 2226	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋 ∙ 𝑌) = (𝑌 · 𝑋))

Syntax hints:   → wi 4   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ‘cfv 5254  (class class class)co 5918  tpos ctpos 6297  Basecbs 12618  ._rcmulr 12696  opp_rcoppr 13563

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-sets 12625  df-mulr 12709  df-oppr 13564

This theorem is referenced by:  crngoppr  13568  opprrng  13573  opprrngbg  13574  opprring  13575  opprringbg  13576  oppr1g  13578  mulgass3  13581  opprunitd  13606  unitmulcl  13609  unitgrp  13612  unitpropdg  13644  rhmopp  13672  opprsubrngg  13707  subrguss  13732  subrgunit  13735  opprdomnbg  13770  isridlrng  13978  isridl  14000  2idlcpblrng  14019