Theorem opprmul 20468

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
opprmul	⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)

Proof of Theorem opprmul

Step	Hyp	Ref	Expression
1		opprval.1	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
2		opprval.2	. . . 4 ⊢ · = (.r‘𝑅)
3		opprval.3	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
4		opprmulfval.4	. . . 4 ⊢ ∙ = (.r‘𝑂)
5	1, 2, 3, 4	opprmulfval 20467	. . 3 ⊢ ∙ = tpos ·
6	5	oveqi 7432	. 2 ⊢ (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌)
7		ovtpos 8243	. 2 ⊢ (𝑋tpos · 𝑌) = (𝑌 · 𝑋)
8	6, 7	eqtri 2788	1 ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)

This proof depends on syntax axioms:   = wceq 1570  ‘cfv 6540  (class class class)co 7419  tpos ctpos 8227  Basecbs 17291  ._rcmulr 17333  opp_rcoppr 20464

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-1cn 11173  ax-addcl 11175

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-nn 12249  df-2 12318  df-3 12319  df-sets 17246  df-slot 17264  df-ndx 17276  df-mulr 17346  df-oppr 20465

This theorem is used by:  crngoppr  20469  opprrng  20473  opprrngb  20474  opprring  20475  opprringb  20476  oppr1  20478  mulgass3  20481  opprunit  20505  unitmulcl  20508  unitgrp  20511  unitpropd  20545  opprirred  20550  irredlmul  20556  rhmopp  20656  opprsubrng  20708  subrguss  20736  subrgunit  20739  opprsubrg  20742  opprdomnb  20865  isdomn4r  20867  isdrng4  20889  isdrng2  20893  isdrngrd  20919  isdrngrdOLD  20921  srngmul  21005  issrngd  21008  rngridlmcl  21392  isridlrng  21394  isridl  21441  2idlcpblrng  21460  psropprmul  22447  invrvald  22883  isunit2  33623  opprlidlabs  33831  opprqusmulr  33837  qsdrngi  33841  ldualsmul  39967  lcdsmul  42434