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Theorem opprmulfval 15722
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 5734 . . . . . . 7  |-  ( .r
`  R )  e. 
_V
42, 3eqeltri 2505 . . . . . 6  |-  .x.  e.  _V
54tposex 6505 . . . . 5  |- tpos  .x.  e.  _V
6 mulrid 13567 . . . . . 6  |-  .r  = Slot  ( .r `  ndx )
76setsid 13500 . . . . 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 15721 . . . . 5  |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
1211fveq2i 5723 . . . 4  |-  ( .r
`  O )  =  ( .r `  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
)
138, 12syl6reqr 2486 . . 3  |-  ( R  e.  _V  ->  ( .r `  O )  = tpos  .x.  )
14 tpos0 6501 . . . . 5  |- tpos  (/)  =  (/)
156str0 13497 . . . . 5  |-  (/)  =  ( .r `  (/) )
1614, 15eqtr2i 2456 . . . 4  |-  ( .r
`  (/) )  = tpos  (/)
17 fvprc 5714 . . . . . 6  |-  ( -.  R  e.  _V  ->  (oppr `  R )  =  (/) )
1810, 17syl5eq 2479 . . . . 5  |-  ( -.  R  e.  _V  ->  O  =  (/) )
1918fveq2d 5724 . . . 4  |-  ( -.  R  e.  _V  ->  ( .r `  O )  =  ( .r `  (/) ) )
20 fvprc 5714 . . . . . 6  |-  ( -.  R  e.  _V  ->  ( .r `  R )  =  (/) )
212, 20syl5eq 2479 . . . . 5  |-  ( -.  R  e.  _V  ->  .x.  =  (/) )
2221tposeqd 6474 . . . 4  |-  ( -.  R  e.  _V  -> tpos  .x.  = tpos 
(/) )
2316, 19, 223eqtr4a 2493 . . 3  |-  ( -.  R  e.  _V  ->  ( .r `  O )  = tpos  .x.  )
2413, 23pm2.61i 158 . 2  |-  ( .r
`  O )  = tpos  .x.
251, 24eqtri 2455 1  |-  .xb  = tpos  .x.
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   <.cop 3809   ` cfv 5446  (class class class)co 6073  tpos ctpos 6470   ndxcnx 13458   sSet csts 13459   Basecbs 13461   .rcmulr 13522  opprcoppr 15719
This theorem is referenced by:  opprmul  15723
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-i2m1 9050  ax-1ne0 9051  ax-rrecex 9054  ax-cnre 9055
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-tpos 6471  df-recs 6625  df-rdg 6660  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-sets 13467  df-mulr 13535  df-oppr 15720
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