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Theorem evlval 19424
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 5883 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( i evalSub  r )  =  ( I evalSub  R
) )
3 fveq2 5541 . . . . . . 7  |-  ( r  =  R  ->  ( Base `  r )  =  ( Base `  R
) )
4 evlval.b . . . . . . 7  |-  B  =  ( Base `  R
)
53, 4syl6eqr 2346 . . . . . 6  |-  ( r  =  R  ->  ( Base `  r )  =  B )
65adantl 452 . . . . 5  |-  ( ( i  =  I  /\  r  =  R )  ->  ( Base `  r
)  =  B )
72, 6fveq12d 5547 . . . 4  |-  ( ( i  =  I  /\  r  =  R )  ->  ( ( i evalSub  r
) `  ( Base `  r ) )  =  ( ( I evalSub  R
) `  B )
)
8 df-evl 16118 . . . 4  |- eval  =  ( i  e.  _V , 
r  e.  _V  |->  ( ( i evalSub  r ) `
 ( Base `  r
) ) )
9 fvex 5555 . . . 4  |-  ( ( I evalSub  R ) `  B
)  e.  _V
107, 8, 9ovmpt2a 5994 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( ( I evalSub  R ) `  B
) )
118reldmmpt2 5971 . . . . . 6  |-  Rel  dom eval
1211ovprc 5901 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  (/) )
13 fv01 5575 . . . . 5  |-  ( (/) `  B )  =  (/)
1412, 13syl6eqr 2346 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I eval  R )  =  ( (/) `  B
) )
15 reldmevls 19417 . . . . . 6  |-  Rel  dom evalSub
1615ovprc 5901 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I evalSub  R )  =  (/) )
1716fveq1d 5543 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( ( I evalSub  R
) `  B )  =  ( (/) `  B
) )
1814, 17eqtr4d 2331 . . 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 2316 1  |-  Q  =  ( ( I evalSub  R
) `  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ` cfv 5271  (class class class)co 5874   Basecbs 13164   evalSub ces 16106   eval cevl 16107
This theorem is referenced by:  evlrhm  19425  evl1sca  19429  evl1var  19431  pf1rcl  19448  mpfpf1  19450  pf1ind  19454  mzpmfp  26928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-evls 16117  df-evl 16118
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