Theorem opprmulg 13248

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 13247	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∙ = tpos · )
6	5	oveqd 5894	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌))
7	6	3ad2ant1 1018	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌))
8		ovtposg 6262	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋tpos · 𝑌) = (𝑌 · 𝑋))
9	8	3adant1 1015	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋tpos · 𝑌) = (𝑌 · 𝑋))
10	7, 9	eqtrd 2210	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋 ∙ 𝑌) = (𝑌 · 𝑋))

Syntax hints:   → wi 4   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ‘cfv 5218  (class class class)co 5877  tpos ctpos 6247  Basecbs 12464  ._rcmulr 12539  opp_rcoppr 13244

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-tpos 6248  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-sets 12471  df-mulr 12552  df-oppr 13245

This theorem is referenced by:  crngoppr  13249  opprring  13254  opprringbg  13255  oppr1g  13257  mulgass3  13259  opprunitd  13284  unitmulcl  13287  unitgrp  13290  unitpropdg  13322  subrguss  13362  subrgunit  13365