Theorem reldmmhp 22045

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 22044	. 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 7475	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1541   ∈ wcel 2110  {crab 3393  Vcvv 3434   ⊆ wss 3900   ↦ cmpt 5170  ^◡ccnv 5613  dom cdm 5614   “ cima 5617  Rel wrel 5619  ‘cfv 6477  (class class class)co 7341   supp csupp 8085   ↑_m cmap 8745  Fincfn 8864  ℕcn 12117  ℕ₀cn0 12373  Basecbs 17112   ↾_s cress 17133  0_gc0g 17335   Σ_g cgsu 17336  ℂ_fldccnfld 21284   mPoly cmpl 21836   mHomP cmhp 22037

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-dm 5624  df-oprab 7345  df-mpo 7346  df-mhp 22044

This theorem is referenced by:  ismhp  22048  mhprcl  22051  mhpmulcl  22057  mhppwdeg  22058  mhpaddcl  22059  mhpinvcl  22060  mhpvscacl  22062  mhpind  42606  evlsmhpvvval  42607  mhphf2  42610  mhphf3  42611