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Theorem evlval 17622
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 6112 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i evalSub  r )  =  ( I evalSub  R
) )
3 fveq2 5703 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 evlval.b . . . . . . 7  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2493 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  B )
65adantl 466 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  r
)  =  B )
72, 6fveq12d 5709 . . . 4  |-  ( ( i  =  I  /\  r  =  R )  ->  ( ( i evalSub  r
) `  ( Base `  r ) )  =  ( ( I evalSub  R
) `  B )
)
8 df-evl 17601 . . . 4  |- eval  =  ( i  e.  _V , 
r  e.  _V  |->  ( ( i evalSub  r ) `
 ( Base `  r
) ) )
9 fvex 5713 . . . 4  |-  ( ( I evalSub  R ) `  B
)  e.  _V
107, 8, 9ovmpt2a 6233 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( ( I evalSub  R ) `  B
) )
118mpt2ndm0 6751 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  (/) )
12 0fv 5735 . . . . 5  |-  ( (/) `  B )  =  (/)
1311, 12syl6eqr 2493 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( (/) `  B
) )
14 reldmevls 17615 . . . . . 6  |-  Rel  dom evalSub
1514ovprc 6130 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I evalSub  R )  =  (/) )
1615fveq1d 5705 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( ( I evalSub  R
) `  B )  =  ( (/) `  B
) )
1713, 16eqtr4d 2478 . . 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 2463 1  |-  Q  =  ( ( I evalSub  R
) `  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2984   (/)c0 3649   ` cfv 5430  (class class class)co 6103   Basecbs 14186   evalSub ces 17598   eval cevl 17599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-evls 17600  df-evl 17601
This theorem is referenced by:  evlrhm  17623  evlsscasrng  17624  evlsvarsrng  17626  evl1fval1lem  17776  evl1sca  17780  evl1var  17782  pf1rcl  17795  mpfpf1  17797  pf1ind  17801  mzpmfp  29095  mzpmfpOLD  29096
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