Theorem opprmulg 13196

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 13195	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∙ = tpos · )
6	5	oveqd 5891	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌))
7	6	3ad2ant1 1018	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌))
8		ovtposg 6259	. . 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 5216  (class class class)co 5874  tpos ctpos 6244  Basecbs 12456  ._rcmulr 12531  opp_rcoppr 13192

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 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-sets 12463  df-mulr 12544  df-oppr 13193

This theorem is referenced by:  crngoppr  13197  opprring  13202  opprringbg  13203  oppr1g  13205  mulgass3  13207  opprunitd  13232  unitmulcl  13235  unitgrp  13238  unitpropdg  13270  subrguss  13317  subrgunit  13320