Theorem reldmmhp 22062

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 22061	. 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 7489	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1541   ∈ wcel 2113  {crab 3397  Vcvv 3438   ⊆ wss 3899   ↦ cmpt 5176  ^◡ccnv 5620  dom cdm 5621   “ cima 5624  Rel wrel 5626  ‘cfv 6489  (class class class)co 7355   supp csupp 8099   ↑_m cmap 8759  Fincfn 8878  ℕcn 12135  ℕ₀cn0 12391  Basecbs 17130   ↾_s cress 17151  0_gc0g 17353   Σ_g cgsu 17354  ℂ_fldccnfld 21301   mPoly cmpl 21853   mHomP cmhp 22054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-dm 5631  df-oprab 7359  df-mpo 7360  df-mhp 22061

This theorem is referenced by:  ismhp  22065  mhprcl  22068  mhpmulcl  22074  mhppwdeg  22075  mhpaddcl  22076  mhpinvcl  22077  mhpvscacl  22079  mhpind  42702  evlsmhpvvval  42703  mhphf2  42706  mhphf3  42707