MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evl1fval1 Structured version   Visualization version   GIF version

Theorem evl1fval1 21713
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 21712 . 2 (𝑅 ∈ V → 𝑄 = (𝑅 evalSub1 𝐵))
4 fvprc 6839 . . . 4 𝑅 ∈ V → (eval1𝑅) = ∅)
51, 4eqtrid 2789 . . 3 𝑅 ∈ V → 𝑄 = ∅)
6 reldmevls1 21699 . . . 4 Rel dom evalSub1
76ovprc1 7401 . . 3 𝑅 ∈ V → (𝑅 evalSub1 𝐵) = ∅)
85, 7eqtr4d 2780 . 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 1542  wcel 2107  Vcvv 3448  c0 4287  cfv 6501  (class class class)co 7362  Basecbs 17090   evalSub1 ces1 21695  eval1ce1 21696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-evls 21498  df-evl 21499  df-evls1 21697  df-evl1 21698
This theorem is referenced by:  evls1scasrng  21721  evls1varsrng  21722  evl1gsumadd  21740  evl1varpw  21743  ressply1evl  32312
  Copyright terms: Public domain W3C validator