Theorem reldmmhp 22080

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 22079	. 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 7546	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1540   ∈ wcel 2109  {crab 3420  Vcvv 3464   ⊆ wss 3931   ↦ cmpt 5206  ^◡ccnv 5658  dom cdm 5659   “ cima 5662  Rel wrel 5664  ‘cfv 6536  (class class class)co 7410   supp csupp 8164   ↑_m cmap 8845  Fincfn 8964  ℕcn 12245  ℕ₀cn0 12506  Basecbs 17233   ↾_s cress 17256  0_gc0g 17458   Σ_g cgsu 17459  ℂ_fldccnfld 21320   mPoly cmpl 21871   mHomP cmhp 22072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-rab 3421  df-v 3466  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 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-dm 5669  df-oprab 7414  df-mpo 7415  df-mhp 22079

This theorem is referenced by:  ismhp  22083  mhprcl  22086  mhpmulcl  22092  mhppwdeg  22093  mhpaddcl  22094  mhpinvcl  22095  mhpvscacl  22097  mhpind  42584  evlsmhpvvval  42585  mhphf2  42588  mhphf3  42589