Theorem reldmmhp 22175

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 22174	. 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 7519	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1554   ∈ wcel 2136  {crab 3408  Vcvv 3448   ⊆ wss 3899   ↦ cmpt 5175  ^◡ccnv 5639  dom cdm 5640   “ cima 5643  Rel wrel 5645  ‘cfv 6510  (class class class)co 7385   supp csupp 8128   ↑_m cmap 8796  Fincfn 8916  ℕcn 12200  ℕ₀cn0 12471  Basecbs 17221   ↾_s cress 17242  0_gc0g 17444   Σ_g cgsu 17445  ℂ_fldccnfld 21397   mPoly cmpl 21931   mHomP cmhp 22171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-xp 5646  df-rel 5647  df-dm 5650  df-oprab 7389  df-mpo 7390  df-mhp 22174

This theorem is referenced by:  ismhp  22178  mhprcl  22181  mhpmulcl  22187  mhppwdeg  22188  mhpaddcl  22189  mhpinvcl  22190  mhpvscacl  22192  mhpind  43124  evlsmhpvvval  43125  mhphf2  43128  mhphf3  43129