Theorem opprmul 20563

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 20562	. . 3 ⊢ ∙ = tpos ·
6	5	oveqi 7431	. 2 ⊢ (𝑋 ∙ 𝑌) = (𝑋tpos · 𝑌)
7		ovtpos 8251	. 2 ⊢ (𝑋tpos · 𝑌) = (𝑌 · 𝑋)
8	6, 7	eqtri 2784	1 ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)

This proof depends on syntax axioms:   = wceq 1570  ‘cfv 6537  (class class class)co 7418  tpos ctpos 8235  Basecbs 17380  ._rcmulr 17422  opp_rcoppr 20559

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-1cn 11251  ax-addcl 11253

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-2nd 8000  df-tpos 8236  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-nn 12329  df-2 12398  df-3 12399  df-sets 17335  df-slot 17353  df-ndx 17365  df-mulr 17435  df-oppr 20560

This theorem is used by:  crngoppr  20564  opprrng  20568  opprrngb  20569  opprring  20570  opprringb  20571  oppr1  20573  mulgass3  20576  opprunit  20600  unitmulcl  20603  unitgrp  20606  unitpropd  20640  opprirred  20645  irredlmul  20651  rhmopp  20752  opprsubrng  20804  subrguss  20832  subrgunit  20835  opprsubrg  20838  opprdomnb  20961  isdomn4r  20963  isdrng4  20985  isdrng2  20990  isdrngrd  21016  isdrngrdOLD  21018  srngmul  21102  issrngd  21105  rngridlmcl  21489  isridlrng  21491  isridl  21538  2idlcpblrng  21558  psropprmul  22548  invrvald  22984  isunit2  33793  opprlidlabs  34002  opprqusmulr  34008  qsdrngi  34012  ldualsmul  40172  lcdsmul  42639