Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  opprval Structured version   Unicode version

Theorem opprval 15729
 Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprval.1
opprval.2
opprval.3 oppr
Assertion
Ref Expression
opprval sSet tpos

Proof of Theorem opprval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 opprval.3 . 2 oppr
2 id 20 . . . . 5
3 fveq2 5728 . . . . . . . 8
4 opprval.2 . . . . . . . 8
53, 4syl6eqr 2486 . . . . . . 7
65tposeqd 6482 . . . . . 6 tpos tpos
76opeq2d 3991 . . . . 5 tpos tpos
82, 7oveq12d 6099 . . . 4 sSet tpos sSet tpos
9 df-oppr 15728 . . . 4 oppr sSet tpos
10 ovex 6106 . . . 4 sSet tpos
118, 9, 10fvmpt 5806 . . 3 oppr sSet tpos
12 fvprc 5722 . . . 4 oppr
13 reldmsets 13491 . . . . 5 sSet
1413ovprc1 6109 . . . 4 sSet tpos
1512, 14eqtr4d 2471 . . 3 oppr sSet tpos
1611, 15pm2.61i 158 . 2 oppr sSet tpos
171, 16eqtri 2456 1 sSet tpos
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1652   wcel 1725  cvv 2956  c0 3628  cop 3817  cfv 5454  (class class class)co 6081  tpos ctpos 6478  cnx 13466   sSet csts 13467  cbs 13469  cmulr 13530  opprcoppr 15727 This theorem is referenced by:  opprmulfval  15730  opprlem  15733 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-sets 13475  df-oppr 15728
 Copyright terms: Public domain W3C validator