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Theorem evlval 19408
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 5867 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i evalSub  r )  =  ( I evalSub  R
) )
3 fveq2 5525 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 evlval.b . . . . . . 7  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2333 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  B )
65adantl 452 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  r
)  =  B )
72, 6fveq12d 5531 . . . 4  |-  ( ( i  =  I  /\  r  =  R )  ->  ( ( i evalSub  r
) `  ( Base `  r ) )  =  ( ( I evalSub  R
) `  B )
)
8 df-evl 16102 . . . 4  |- eval  =  ( i  e.  _V , 
r  e.  _V  |->  ( ( i evalSub  r ) `
 ( Base `  r
) ) )
9 fvex 5539 . . . 4  |-  ( ( I evalSub  R ) `  B
)  e.  _V
107, 8, 9ovmpt2a 5978 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( ( I evalSub  R ) `  B
) )
118reldmmpt2 5955 . . . . . 6  |-  Rel  dom eval
1211ovprc 5885 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  (/) )
13 fv01 5559 . . . . 5  |-  ( (/) `  B )  =  (/)
1412, 13syl6eqr 2333 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( (/) `  B
) )
15 reldmevls 19401 . . . . . 6  |-  Rel  dom evalSub
1615ovprc 5885 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I evalSub  R )  =  (/) )
1716fveq1d 5527 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( ( I evalSub  R
) `  B )  =  ( (/) `  B
) )
1814, 17eqtr4d 2318 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( ( I evalSub  R ) `  B
) )
1910, 18pm2.61i 156 . 2  |-  ( I eval 
R )  =  ( ( I evalSub  R ) `
 B )
201, 19eqtri 2303 1  |-  Q  =  ( ( I evalSub  R
) `  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ` cfv 5255  (class class class)co 5858   Basecbs 13148   evalSub ces 16090   eval cevl 16091
This theorem is referenced by:  evlrhm  19409  evl1sca  19413  evl1var  19415  pf1rcl  19432  mpfpf1  19434  pf1ind  19438  mzpmfp  26825
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-evls 16101  df-evl 16102
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