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Theorem opprmulfval 17469
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 5858 . . . . . . 7  |-  ( .r
`  R )  e. 
_V
42, 3eqeltri 2538 . . . . . 6  |-  .x.  e.  _V
54tposex 6981 . . . . 5  |- tpos  .x.  e.  _V
6 mulrid 14834 . . . . . 6  |-  .r  = Slot  ( .r `  ndx )
76setsid 14759 . . . . 5  |-  ( ( R  e.  _V  /\ tpos  .x. 
e.  _V )  -> tpos  .x.  =  ( .r `  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
) )
85, 7mpan2 669 . . . 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 17468 . . . . 5  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
1211fveq2i 5851 . . . 4  |-  ( .r
`  O )  =  ( .r `  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
138, 12syl6reqr 2514 . . 3  |-  ( R  e.  _V  ->  ( .r `  O )  = tpos  .x.  )
14 tpos0 6977 . . . . 5  |- tpos  (/)  =  (/)
156str0 14756 . . . . 5  |-  (/)  =  ( .r `  (/) )
1614, 15eqtr2i 2484 . . . 4  |-  ( .r
`  (/) )  = tpos  (/)
17 fvprc 5842 . . . . . 6  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  (/) )
1810, 17syl5eq 2507 . . . . 5  |-  ( -.  R  e.  _V  ->  O  =  (/) )
1918fveq2d 5852 . . . 4  |-  ( -.  R  e.  _V  ->  ( .r `  O )  =  ( .r `  (/) ) )
20 fvprc 5842 . . . . . 6  |-  ( -.  R  e.  _V  ->  ( .r `  R )  =  (/) )
212, 20syl5eq 2507 . . . . 5  |-  ( -.  R  e.  _V  ->  .x.  =  (/) )
2221tposeqd 6950 . . . 4  |-  ( -.  R  e.  _V  -> tpos  .x.  = tpos 
(/) )
2316, 19, 223eqtr4a 2521 . . 3  |-  ( -.  R  e.  _V  ->  ( .r `  O )  = tpos  .x.  )
2413, 23pm2.61i 164 . 2  |-  ( .r
`  O )  = tpos  .x.
251, 24eqtri 2483 1  |-  .xb  = tpos  .x.
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   <.cop 4022   ` cfv 5570  (class class class)co 6270  tpos ctpos 6946   ndxcnx 14713   sSet csts 14714   Basecbs 14716   .rcmulr 14785  opprcoppr 17466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-sets 14722  df-mulr 14798  df-oppr 17467
This theorem is referenced by:  opprmul  17470
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