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Theorem opprmulfval 15407
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 5539 . . . . . . 7  |-  ( .r
`  R )  e. 
_V
42, 3eqeltri 2353 . . . . . 6  |-  .x.  e.  _V
54tposex 6268 . . . . 5  |- tpos  .x.  e.  _V
6 mulrid 13254 . . . . . 6  |-  .r  = Slot  ( .r `  ndx )
76setsid 13187 . . . . 5  |-  ( ( R  e.  _V  /\ tpos  .x. 
e.  _V )  -> tpos  .x.  =  ( .r `  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
) )
85, 7mpan2 652 . . . 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 15406 . . . . 5  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
1211fveq2i 5528 . . . 4  |-  ( .r
`  O )  =  ( .r `  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
138, 12syl6reqr 2334 . . 3  |-  ( R  e.  _V  ->  ( .r `  O )  = tpos  .x.  )
14 tpos0 6264 . . . . 5  |- tpos  (/)  =  (/)
156str0 13184 . . . . 5  |-  (/)  =  ( .r `  (/) )
1614, 15eqtr2i 2304 . . . 4  |-  ( .r
`  (/) )  = tpos  (/)
17 fvprc 5519 . . . . . 6  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  (/) )
1810, 17syl5eq 2327 . . . . 5  |-  ( -.  R  e.  _V  ->  O  =  (/) )
1918fveq2d 5529 . . . 4  |-  ( -.  R  e.  _V  ->  ( .r `  O )  =  ( .r `  (/) ) )
20 fvprc 5519 . . . . . 6  |-  ( -.  R  e.  _V  ->  ( .r `  R )  =  (/) )
212, 20syl5eq 2327 . . . . 5  |-  ( -.  R  e.  _V  ->  .x.  =  (/) )
22 tposeq 6236 . . . . 5  |-  (  .x.  =  (/)  -> tpos  .x.  = tpos  (/) )
2321, 22syl 15 . . . 4  |-  ( -.  R  e.  _V  -> tpos  .x.  = tpos 
(/) )
2416, 19, 233eqtr4a 2341 . . 3  |-  ( -.  R  e.  _V  ->  ( .r `  O )  = tpos  .x.  )
2513, 24pm2.61i 156 . 2  |-  ( .r
`  O )  = tpos  .x.
261, 25eqtri 2303 1  |-  .xb  = tpos  .x.
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   <.cop 3643   ` cfv 5255  (class class class)co 5858  tpos ctpos 6233   ndxcnx 13145   sSet csts 13146   Basecbs 13148   .rcmulr 13209  opprcoppr 15404
This theorem is referenced by:  opprmul  15408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tpos 6234  df-recs 6388  df-rdg 6423  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-sets 13154  df-mulr 13222  df-oppr 15405
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