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Theorem opprmulfval 15658
Description: Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprval.1  |-  B  =  ( Base `  R
)
opprval.2  |-  .x.  =  ( .r `  R )
opprval.3  |-  O  =  (oppr
`  R )
opprmulfval.4  |-  .xb  =  ( .r `  O )
Assertion
Ref Expression
opprmulfval  |-  .xb  = tpos  .x.

Proof of Theorem opprmulfval
StepHypRef Expression
1 opprmulfval.4 . 2  |-  .xb  =  ( .r `  O )
2 opprval.2 . . . . . . 7  |-  .x.  =  ( .r `  R )
3 fvex 5683 . . . . . . 7  |-  ( .r
`  R )  e. 
_V
42, 3eqeltri 2458 . . . . . 6  |-  .x.  e.  _V
54tposex 6450 . . . . 5  |- tpos  .x.  e.  _V
6 mulrid 13503 . . . . . 6  |-  .r  = Slot  ( .r `  ndx )
76setsid 13436 . . . . 5  |-  ( ( R  e.  _V  /\ tpos  .x. 
e.  _V )  -> tpos  .x.  =  ( .r `  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
) )
85, 7mpan2 653 . . . 4  |-  ( R  e.  _V  -> tpos  .x.  =  ( .r `  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
) )
9 opprval.1 . . . . . 6  |-  B  =  ( Base `  R
)
10 opprval.3 . . . . . 6  |-  O  =  (oppr
`  R )
119, 2, 10opprval 15657 . . . . 5  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
1211fveq2i 5672 . . . 4  |-  ( .r
`  O )  =  ( .r `  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
138, 12syl6reqr 2439 . . 3  |-  ( R  e.  _V  ->  ( .r `  O )  = tpos  .x.  )
14 tpos0 6446 . . . . 5  |- tpos  (/)  =  (/)
156str0 13433 . . . . 5  |-  (/)  =  ( .r `  (/) )
1614, 15eqtr2i 2409 . . . 4  |-  ( .r
`  (/) )  = tpos  (/)
17 fvprc 5663 . . . . . 6  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  (/) )
1810, 17syl5eq 2432 . . . . 5  |-  ( -.  R  e.  _V  ->  O  =  (/) )
1918fveq2d 5673 . . . 4  |-  ( -.  R  e.  _V  ->  ( .r `  O )  =  ( .r `  (/) ) )
20 fvprc 5663 . . . . . 6  |-  ( -.  R  e.  _V  ->  ( .r `  R )  =  (/) )
212, 20syl5eq 2432 . . . . 5  |-  ( -.  R  e.  _V  ->  .x.  =  (/) )
2221tposeqd 6419 . . . 4  |-  ( -.  R  e.  _V  -> tpos  .x.  = tpos 
(/) )
2316, 19, 223eqtr4a 2446 . . 3  |-  ( -.  R  e.  _V  ->  ( .r `  O )  = tpos  .x.  )
2413, 23pm2.61i 158 . 2  |-  ( .r
`  O )  = tpos  .x.
251, 24eqtri 2408 1  |-  .xb  = tpos  .x.
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1717   _Vcvv 2900   (/)c0 3572   <.cop 3761   ` cfv 5395  (class class class)co 6021  tpos ctpos 6415   ndxcnx 13394   sSet csts 13395   Basecbs 13397   .rcmulr 13458  opprcoppr 15655
This theorem is referenced by:  opprmul  15659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-i2m1 8992  ax-1ne0 8993  ax-rrecex 8996  ax-cnre 8997
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-tpos 6416  df-recs 6570  df-rdg 6605  df-nn 9934  df-2 9991  df-3 9992  df-ndx 13400  df-slot 13401  df-sets 13403  df-mulr 13471  df-oppr 15656
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