Theorem opprmulg 14087

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 14086	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∙ = tpos · )
6	5	oveqd 6035	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌))
7	6	3ad2ant1 1044	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌))
8		ovtposg 6425	. . 3 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋tpos · 𝑌) = (𝑌 · 𝑋))
9	8	3adant1 1041	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋tpos · 𝑌) = (𝑌 · 𝑋))
10	7, 9	eqtrd 2264	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑈) → (𝑋 ∙ 𝑌) = (𝑌 · 𝑋))

Syntax hints:   → wi 4   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ‘cfv 5326  (class class class)co 6018  tpos ctpos 6410  Basecbs 13084  ._rcmulr 13163  opp_rcoppr 14083

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-tpos 6411  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13087  df-slot 13088  df-sets 13091  df-mulr 13176  df-oppr 14084

This theorem is referenced by:  crngoppr  14088  opprrng  14093  opprrngbg  14094  opprring  14095  opprringbg  14096  oppr1g  14098  mulgass3  14101  opprunitd  14127  unitmulcl  14130  unitgrp  14133  unitpropdg  14165  rhmopp  14193  opprsubrngg  14228  subrguss  14253  subrgunit  14256  opprdomnbg  14291  isridlrng  14499  isridl  14521  2idlcpblrng  14540