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Theorem evlval 19945
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 6090 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i evalSub  r )  =  ( I evalSub  R
) )
3 fveq2 5728 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 evlval.b . . . . . . 7  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2486 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  B )
65adantl 453 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  r
)  =  B )
72, 6fveq12d 5734 . . . 4  |-  ( ( i  =  I  /\  r  =  R )  ->  ( ( i evalSub  r
) `  ( Base `  r ) )  =  ( ( I evalSub  R
) `  B )
)
8 df-evl 16421 . . . 4  |- eval  =  ( i  e.  _V , 
r  e.  _V  |->  ( ( i evalSub  r ) `
 ( Base `  r
) ) )
9 fvex 5742 . . . 4  |-  ( ( I evalSub  R ) `  B
)  e.  _V
107, 8, 9ovmpt2a 6204 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( ( I evalSub  R ) `  B
) )
118mpt2ndm0 6473 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  (/) )
12 fv01 5763 . . . . 5  |-  ( (/) `  B )  =  (/)
1311, 12syl6eqr 2486 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( (/) `  B
) )
14 reldmevls 19938 . . . . . 6  |-  Rel  dom evalSub
1514ovprc 6108 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I evalSub  R )  =  (/) )
1615fveq1d 5730 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( ( I evalSub  R
) `  B )  =  ( (/) `  B
) )
1713, 16eqtr4d 2471 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( ( I evalSub  R ) `  B
) )
1810, 17pm2.61i 158 . 2  |-  ( I eval 
R )  =  ( ( I evalSub  R ) `
 B )
191, 18eqtri 2456 1  |-  Q  =  ( ( I evalSub  R
) `  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   ` cfv 5454  (class class class)co 6081   Basecbs 13469   evalSub ces 16409   eval cevl 16410
This theorem is referenced by:  evlrhm  19946  evl1sca  19950  evl1var  19952  pf1rcl  19969  mpfpf1  19971  pf1ind  19975  mzpmfp  26804
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-evls 16420  df-evl 16421
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