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Theorem evl1fval1 19676
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 = (eval1𝑅)
evl1fval1.b 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
evl1fval1 𝑄 = (𝑅 evalSub1 𝐵)

Proof of Theorem evl1fval1
StepHypRef Expression
1 evl1fval1.q . . 3 𝑄 = (eval1𝑅)
2 evl1fval1.b . . 3 𝐵 = (Base‘𝑅)
31, 2evl1fval1lem 19675 . 2 (𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵))
4 fvprc 6172 . . . 4 𝑅 ∈ V → (eval1𝑅) = ∅)
51, 4syl5eq 2666 . . 3 𝑅 ∈ V → 𝑄 = ∅)
6 reldmevls1 19663 . . . 4 Rel dom evalSub1
76ovprc1 6669 . . 3 𝑅 ∈ V → (𝑅 evalSub1 𝐵) = ∅)
85, 7eqtr4d 2657 . 2 𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵))
93, 8pm2.61i 176 1 𝑄 = (𝑅 evalSub1 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  cfv 5876  (class class class)co 6635  Basecbs 15838   evalSub1 ces1 19659  eval1ce1 19660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-evls 19487  df-evl 19488  df-evls1 19661  df-evl1 19662
This theorem is referenced by:  evls1scasrng  19684  evls1varsrng  19685  evl1gsumadd  19703  evl1varpw  19706
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