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Theorem evl1fval1 22205
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 22204 . 2 (𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵))
4 fvprc 6877 . . . 4 𝑅 ∈ V → (eval1𝑅) = ∅)
51, 4eqtrid 2778 . . 3 𝑅 ∈ V → 𝑄 = ∅)
6 reldmevls1 22191 . . . 4 Rel dom evalSub1
76ovprc1 7444 . . 3 𝑅 ∈ V → (𝑅 evalSub1 𝐵) = ∅)
85, 7eqtr4d 2769 . 2 𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵))
93, 8pm2.61i 182 1 𝑄 = (𝑅 evalSub1 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3468  c0 4317  cfv 6537  (class class class)co 7405  Basecbs 17153   evalSub1 ces1 22187  eval1ce1 22188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-evls 21977  df-evl 21978  df-evls1 22189  df-evl1 22190
This theorem is referenced by:  evls1scasrng  22213  evls1varsrng  22214  evl1gsumadd  22232  evl1varpw  22235  ressply1evl  33156
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