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Theorem evlval 18004
Description: Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
evlval.q  |-  Q  =  ( I eval  R )
evlval.b  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
evlval  |-  Q  =  ( ( I evalSub  R
) `  B )

Proof of Theorem evlval
Dummy variables  i 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlval.q . 2  |-  Q  =  ( I eval  R )
2 oveq12 6294 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i evalSub  r )  =  ( I evalSub  R
) )
3 fveq2 5866 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 evlval.b . . . . . . 7  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2526 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  B )
65adantl 466 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  r
)  =  B )
72, 6fveq12d 5872 . . . 4  |-  ( ( i  =  I  /\  r  =  R )  ->  ( ( i evalSub  r
) `  ( Base `  r ) )  =  ( ( I evalSub  R
) `  B )
)
8 df-evl 17983 . . . 4  |- eval  =  ( i  e.  _V , 
r  e.  _V  |->  ( ( i evalSub  r ) `
 ( Base `  r
) ) )
9 fvex 5876 . . . 4  |-  ( ( I evalSub  R ) `  B
)  e.  _V
107, 8, 9ovmpt2a 6418 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( ( I evalSub  R ) `  B
) )
118mpt2ndm0 6501 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  (/) )
12 0fv 5899 . . . . 5  |-  ( (/) `  B )  =  (/)
1311, 12syl6eqr 2526 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( (/) `  B
) )
14 reldmevls 17997 . . . . . 6  |-  Rel  dom evalSub
1514ovprc 6312 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I evalSub  R )  =  (/) )
1615fveq1d 5868 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( ( I evalSub  R
) `  B )  =  ( (/) `  B
) )
1713, 16eqtr4d 2511 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( ( I evalSub  R ) `  B
) )
1810, 17pm2.61i 164 . 2  |-  ( I eval 
R )  =  ( ( I evalSub  R ) `
 B )
191, 18eqtri 2496 1  |-  Q  =  ( ( I evalSub  R
) `  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   ` cfv 5588  (class class class)co 6285   Basecbs 14493   evalSub ces 17980   eval cevl 17981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-evls 17982  df-evl 17983
This theorem is referenced by:  evlrhm  18005  evlsscasrng  18006  evlsvarsrng  18008  evl1fval1lem  18177  evl1sca  18181  evl1var  18183  pf1rcl  18196  mpfpf1  18198  pf1ind  18202  mzpmfp  30510  mzpmfpOLD  30511
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