Theorem reldmmhp 22024

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 22023	. 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 7523	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ⊆ wss 3914   ↦ cmpt 5188  ^◡ccnv 5637  dom cdm 5638   “ cima 5641  Rel wrel 5643  ‘cfv 6511  (class class class)co 7387   supp csupp 8139   ↑_m cmap 8799  Fincfn 8918  ℕcn 12186  ℕ₀cn0 12442  Basecbs 17179   ↾_s cress 17200  0_gc0g 17402   Σ_g cgsu 17403  ℂ_fldccnfld 21264   mPoly cmpl 21815   mHomP cmhp 22016

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dm 5648  df-oprab 7391  df-mpo 7392  df-mhp 22023

This theorem is referenced by:  ismhp  22027  mhprcl  22030  mhpmulcl  22036  mhppwdeg  22037  mhpaddcl  22038  mhpinvcl  22039  mhpvscacl  22041  mhpind  42582  evlsmhpvvval  42583  mhphf2  42586  mhphf3  42587