Theorem reldmmhp 22279

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 22278	. 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 7544	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1568   ∈ wcel 2141  {crab 3414  Vcvv 3453   ⊆ wss 3904   ↦ cmpt 5191  ^◡ccnv 5660  dom cdm 5661   “ cima 5664  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410   supp csupp 8155   ↑_m cmap 8823  Fincfn 8942  ℕcn 12232  ℕ₀cn0 12503  Basecbs 17268   ↾_s cress 17289  0_gc0g 17491   Σ_g cgsu 17492  ℂ_fldccnfld 21501   mPoly cmpl 22035   mHomP cmhp 22275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-rel 5668  df-dm 5671  df-oprab 7414  df-mpo 7415  df-mhp 22278

This theorem is referenced by:  ismhp  22282  mhprcl  22285  mhpmulcl  22291  mhppwdeg  22292  mhpaddcl  22293  mhpinvcl  22294  mhpvscacl  22296  mhpind  43296  evlsmhpvvval  43297  mhphf2  43300  mhphf3  43301