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Theorem evl1fval1 22170
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 22169 . 2 (𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵))
4 fvprc 6883 . . . 4 𝑅 ∈ V → (eval1𝑅) = ∅)
51, 4eqtrid 2783 . . 3 𝑅 ∈ V → 𝑄 = ∅)
6 reldmevls1 22156 . . . 4 Rel dom evalSub1
76ovprc1 7451 . . 3 𝑅 ∈ V → (𝑅 evalSub1 𝐵) = ∅)
85, 7eqtr4d 2774 . 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 1540  wcel 2105  Vcvv 3473  c0 4322  cfv 6543  (class class class)co 7412  Basecbs 17151   evalSub1 ces1 22152  eval1ce1 22153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-evls 21946  df-evl 21947  df-evls1 22154  df-evl1 22155
This theorem is referenced by:  evls1scasrng  22178  evls1varsrng  22179  evl1gsumadd  22197  evl1varpw  22200  ressply1evl  33087
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