Theorem reldmmhp 22112

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 22111	. 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 7492	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1542   ∈ wcel 2114  {crab 3390  Vcvv 3430   ⊆ wss 3890   ↦ cmpt 5167  ^◡ccnv 5621  dom cdm 5622   “ cima 5625  Rel wrel 5627  ‘cfv 6490  (class class class)co 7358   supp csupp 8101   ↑_m cmap 8764  Fincfn 8884  ℕcn 12163  ℕ₀cn0 12426  Basecbs 17168   ↾_s cress 17189  0_gc0g 17391   Σ_g cgsu 17392  ℂ_fldccnfld 21342   mPoly cmpl 21894   mHomP cmhp 22104

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 5231  ax-pr 5368

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5628  df-rel 5629  df-dm 5632  df-oprab 7362  df-mpo 7363  df-mhp 22111

This theorem is referenced by:  ismhp  22115  mhprcl  22118  mhpmulcl  22124  mhppwdeg  22125  mhpaddcl  22126  mhpinvcl  22127  mhpvscacl  22129  mhpind  43038  evlsmhpvvval  43039  mhphf2  43042  mhphf3  43043