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Theorem reldmmpl 19191
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 19122 . 2 mPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑖 mPwSer 𝑟) / 𝑠(𝑠s {𝑓 ∈ (Base‘𝑠) ∣ 𝑓 finSupp (0g𝑟)}))
21reldmmpt2 6644 1 Rel dom mPoly
Colors of variables: wff setvar class
Syntax hints:  {crab 2896  Vcvv 3169  csb 3495   class class class wbr 4574  dom cdm 5025  Rel wrel 5030  cfv 5787  (class class class)co 6524   finSupp cfsupp 8132  Basecbs 15638  s cress 15639  0gc0g 15866   mPwSer cmps 19115   mPoly cmpl 19117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-br 4575  df-opab 4635  df-xp 5031  df-rel 5032  df-dm 5035  df-oprab 6528  df-mpt2 6529  df-mpl 19122
This theorem is referenced by:  mplval  19192  mplrcl  19254  mplbaspropd  19371  ply1ascl  19392  mdegfval  23540  mdegcl  23547
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