Theorem reldmmhp 22031

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 22030	. 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 7526	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1540   ∈ wcel 2109  {crab 3408  Vcvv 3450   ⊆ wss 3917   ↦ cmpt 5191  ^◡ccnv 5640  dom cdm 5641   “ cima 5644  Rel wrel 5646  ‘cfv 6514  (class class class)co 7390   supp csupp 8142   ↑_m cmap 8802  Fincfn 8921  ℕcn 12193  ℕ₀cn0 12449  Basecbs 17186   ↾_s cress 17207  0_gc0g 17409   Σ_g cgsu 17410  ℂ_fldccnfld 21271   mPoly cmpl 21822   mHomP cmhp 22023

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-dm 5651  df-oprab 7394  df-mpo 7395  df-mhp 22030

This theorem is referenced by:  ismhp  22034  mhprcl  22037  mhpmulcl  22043  mhppwdeg  22044  mhpaddcl  22045  mhpinvcl  22046  mhpvscacl  22048  mhpind  42589  evlsmhpvvval  42590  mhphf2  42593  mhphf3  42594