Theorem reldmmhp 22092

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 22091	. 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 7502	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442   ⊆ wss 3903   ↦ cmpt 5181  ^◡ccnv 5631  dom cdm 5632   “ cima 5635  Rel wrel 5637  ‘cfv 6500  (class class class)co 7368   supp csupp 8112   ↑_m cmap 8775  Fincfn 8895  ℕcn 12157  ℕ₀cn0 12413  Basecbs 17148   ↾_s cress 17169  0_gc0g 17371   Σ_g cgsu 17372  ℂ_fldccnfld 21321   mPoly cmpl 21874   mHomP cmhp 22084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-dm 5642  df-oprab 7372  df-mpo 7373  df-mhp 22091

This theorem is referenced by:  ismhp  22095  mhprcl  22098  mhpmulcl  22104  mhppwdeg  22105  mhpaddcl  22106  mhpinvcl  22107  mhpvscacl  22109  mhpind  42941  evlsmhpvvval  42942  mhphf2  42945  mhphf3  42946