Theorem reldmmhp 22260

Description: The domain of the homogeneous polynomial operator is a relation. (Contributed by SN, 18-May-2025.)

Assertion

Ref	Expression
reldmmhp	⊢ Rel dom mHomP

Proof of Theorem reldmmhp

Dummy variables 𝑓 𝑔 ℎ 𝑖 𝑛 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mhp 22259	. 2 ⊢ mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
2	1	reldmmpo 7534	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1563   ∈ wcel 2145  {crab 3417  Vcvv 3457   ⊆ wss 3907   ↦ cmpt 5186  ^◡ccnv 5651  dom cdm 5652   “ cima 5655  Rel wrel 5657  ‘cfv 6525  (class class class)co 7400   supp csupp 8144   ↑_m cmap 8812  Fincfn 8931  ℕcn 12224  ℕ₀cn0 12495  Basecbs 17259   ↾_s cress 17280  0_gc0g 17482   Σ_g cgsu 17483  ℂ_fldccnfld 21482   mPoly cmpl 22016   mHomP cmhp 22256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-dm 5662  df-oprab 7404  df-mpo 7405  df-mhp 22259

This theorem is referenced by:  ismhp  22263  mhprcl  22266  mhpmulcl  22272  mhppwdeg  22273  mhpaddcl  22274  mhpinvcl  22275  mhpvscacl  22277  mhpind  43188  evlsmhpvvval  43189  mhphf2  43192  mhphf3  43193