Theorem reldmmhp 22084

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 22083	. 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 7494	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1542   ∈ wcel 2114  {crab 3400  Vcvv 3441   ⊆ wss 3902   ↦ cmpt 5180  ^◡ccnv 5624  dom cdm 5625   “ cima 5628  Rel wrel 5630  ‘cfv 6493  (class class class)co 7360   supp csupp 8104   ↑_m cmap 8767  Fincfn 8887  ℕcn 12149  ℕ₀cn0 12405  Basecbs 17140   ↾_s cress 17161  0_gc0g 17363   Σ_g cgsu 17364  ℂ_fldccnfld 21313   mPoly cmpl 21866   mHomP cmhp 22076

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 5242  ax-nul 5252  ax-pr 5378

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 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-dm 5635  df-oprab 7364  df-mpo 7365  df-mhp 22083

This theorem is referenced by:  ismhp  22087  mhprcl  22090  mhpmulcl  22096  mhppwdeg  22097  mhpaddcl  22098  mhpinvcl  22099  mhpvscacl  22101  mhpind  42873  evlsmhpvvval  42874  mhphf2  42877  mhphf3  42878