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Theorem reldmmpl 22008
Description: The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
reldmmpl Rel dom mPoly

Proof of Theorem reldmmpl
Dummy variables 𝑓 𝑖 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpl 21931 . 2 mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}))
21reldmmpo 7567 1 Rel dom mPoly
Colors of variables: wff setvar class
Syntax hints:  {crab 3436  Vcvv 3480  csb 3899   class class class wbr 5143  dom cdm 5685  Rel wrel 5690  cfv 6561  (class class class)co 7431   finSupp cfsupp 9401  Basecbs 17247  s cress 17274  0gc0g 17484   mPwSer cmps 21924   mPoly cmpl 21926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dm 5695  df-oprab 7435  df-mpo 7436  df-mpl 21931
This theorem is referenced by:  mplval  22009  mplrcl  22014  selvval  22139  ismhp  22144  psdmplcl  22166  mplbaspropd  22238  ply1ascl  22261  mdegfval  26101
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