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Theorem evl1fval1 19516
 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 19515 . 2 (𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵))
4 fvprc 6097 . . . 4 𝑅 ∈ V → (eval1𝑅) = ∅)
51, 4syl5eq 2656 . . 3 𝑅 ∈ V → 𝑄 = ∅)
6 reldmevls1 19503 . . . 4 Rel dom evalSub1
76ovprc1 6582 . . 3 𝑅 ∈ V → (𝑅 evalSub1 𝐵) = ∅)
85, 7eqtr4d 2647 . 2 𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵))
93, 8pm2.61i 175 1 𝑄 = (𝑅 evalSub1 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   evalSub1 ces1 19499  eval1ce1 19500 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-evls 19327  df-evl 19328  df-evls1 19501  df-evl1 19502 This theorem is referenced by:  evls1scasrng  19524  evls1varsrng  19525  evl1gsumadd  19543  evl1varpw  19546
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