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Theorem opprmulfval 15423
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 5555 . . . . . . 7  |-  ( .r
`  R )  e. 
_V
42, 3eqeltri 2366 . . . . . 6  |-  .x.  e.  _V
54tposex 6284 . . . . 5  |- tpos  .x.  e.  _V
6 mulrid 13270 . . . . . 6  |-  .r  = Slot  ( .r `  ndx )
76setsid 13203 . . . . 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 15422 . . . . 5  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
1211fveq2i 5544 . . . 4  |-  ( .r
`  O )  =  ( .r `  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
138, 12syl6reqr 2347 . . 3  |-  ( R  e.  _V  ->  ( .r `  O )  = tpos  .x.  )
14 tpos0 6280 . . . . 5  |- tpos  (/)  =  (/)
156str0 13200 . . . . 5  |-  (/)  =  ( .r `  (/) )
1614, 15eqtr2i 2317 . . . 4  |-  ( .r
`  (/) )  = tpos  (/)
17 fvprc 5535 . . . . . 6  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  (/) )
1810, 17syl5eq 2340 . . . . 5  |-  ( -.  R  e.  _V  ->  O  =  (/) )
1918fveq2d 5545 . . . 4  |-  ( -.  R  e.  _V  ->  ( .r `  O )  =  ( .r `  (/) ) )
20 fvprc 5535 . . . . . 6  |-  ( -.  R  e.  _V  ->  ( .r `  R )  =  (/) )
212, 20syl5eq 2340 . . . . 5  |-  ( -.  R  e.  _V  ->  .x.  =  (/) )
22 tposeq 6252 . . . . 5  |-  (  .x.  =  (/)  -> tpos  .x.  = tpos  (/) )
2321, 22syl 15 . . . 4  |-  ( -.  R  e.  _V  -> tpos  .x.  = tpos 
(/) )
2416, 19, 233eqtr4a 2354 . . 3  |-  ( -.  R  e.  _V  ->  ( .r `  O )  = tpos  .x.  )
2513, 24pm2.61i 156 . 2  |-  ( .r
`  O )  = tpos  .x.
261, 25eqtri 2316 1  |-  .xb  = tpos  .x.
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   <.cop 3656   ` cfv 5271  (class class class)co 5874  tpos ctpos 6249   ndxcnx 13161   sSet csts 13162   Basecbs 13164   .rcmulr 13225  opprcoppr 15420
This theorem is referenced by:  opprmul  15424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-recs 6404  df-rdg 6439  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-sets 13170  df-mulr 13238  df-oppr 15421
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