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Theorem evl1fval1 18237
Description: Value of the simple/same ring evaluation map function for univariate polynomials. (Contributed by AV, 11-Sep-2019.)
Hypotheses
Ref Expression
evl1fval1.q  |-  Q  =  (eval1 `  R )
evl1fval1.b  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
evl1fval1  |-  Q  =  ( R evalSub1  B )

Proof of Theorem evl1fval1
StepHypRef Expression
1 evl1fval1.q . . 3  |-  Q  =  (eval1 `  R )
2 evl1fval1.b . . 3  |-  B  =  ( Base `  R
)
31, 2evl1fval1lem 18236 . 2  |-  ( R  e.  _V  ->  Q  =  ( R evalSub1  B ) )
4 fvprc 5866 . . . 4  |-  ( -.  R  e.  _V  ->  (eval1 `  R )  =  (/) )
51, 4syl5eq 2520 . . 3  |-  ( -.  R  e.  _V  ->  Q  =  (/) )
6 reldmevls1 18224 . . . 4  |-  Rel  dom evalSub1
76ovprc1 6323 . . 3  |-  ( -.  R  e.  _V  ->  ( R evalSub1  B )  =  (/) )
85, 7eqtr4d 2511 . 2  |-  ( -.  R  e.  _V  ->  Q  =  ( R evalSub1  B ) )
93, 8pm2.61i 164 1  |-  Q  =  ( R evalSub1  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   ` cfv 5594  (class class class)co 6295   Basecbs 14507   evalSub1 ces1 18220  eval1ce1 18221
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-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-evls 18041  df-evl 18042  df-evls1 18222  df-evl1 18223
This theorem is referenced by:  evls1scasrng  18245  evls1varsrng  18246  evl1gsumadd  18264  evl1varpw  18267
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