Theorem reldmmhp 22090

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 22089	. 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 7549	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1539   ∈ wcel 2107  {crab 3419  Vcvv 3463   ⊆ wss 3931   ↦ cmpt 5205  ^◡ccnv 5664  dom cdm 5665   “ cima 5668  Rel wrel 5670  ‘cfv 6541  (class class class)co 7413   supp csupp 8167   ↑_m cmap 8848  Fincfn 8967  ℕcn 12248  ℕ₀cn0 12509  Basecbs 17230   ↾_s cress 17253  0_gc0g 17456   Σ_g cgsu 17457  ℂ_fldccnfld 21327   mPoly cmpl 21881   mHomP cmhp 22082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-dm 5675  df-oprab 7417  df-mpo 7418  df-mhp 22089

This theorem is referenced by:  ismhp  22093  mhprcl  22096  mhpmulcl  22102  mhppwdeg  22103  mhpaddcl  22104  mhpinvcl  22105  mhpvscacl  22107  mhpind  42583  evlsmhpvvval  42584  mhphf2  42587  mhphf3  42588