Theorem reldmmhp 22132

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 22131	. 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 7497	1 ⊢ Rel dom mHomP

Syntax hints:   = wceq 1547   ∈ wcel 2119  {crab 3392  Vcvv 3432   ⊆ wss 3890   ↦ cmpt 5160  ^◡ccnv 5624  dom cdm 5625   “ cima 5628  Rel wrel 5630  ‘cfv 6492  (class class class)co 7363   supp csupp 8107   ↑_m cmap 8770  Fincfn 8890  ℕcn 12172  ℕ₀cn0 12435  Basecbs 17177   ↾_s cress 17198  0_gc0g 17400   Σ_g cgsu 17401  ℂ_fldccnfld 21354   mPoly cmpl 21888   mHomP cmhp 22128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-dm 5635  df-oprab 7367  df-mpo 7368  df-mhp 22131

This theorem is referenced by:  ismhp  22135  mhprcl  22138  mhpmulcl  22144  mhppwdeg  22145  mhpaddcl  22146  mhpinvcl  22147  mhpvscacl  22149  mhpind  43045  evlsmhpvvval  43046  mhphf2  43049  mhphf3  43050