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Theorem mhpfval 20789
 Description: Value of the "homogeneous polynomial" function. (Contributed by Steven Nguyen, 25-Aug-2023.)
Hypotheses
Ref Expression
mhpfval.h 𝐻 = (𝐼 mHomP 𝑅)
mhpfval.p 𝑃 = (𝐼 mPoly 𝑅)
mhpfval.b 𝐵 = (Base‘𝑃)
mhpfval.0 0 = (0g𝑅)
mhpfval.d 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
mhpfval.i (𝜑𝐼𝑉)
mhpfval.r (𝜑𝑅𝑊)
Assertion
Ref Expression
mhpfval (𝜑𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
Distinct variable groups:   𝑓,𝑔,,𝑛   𝑓,𝐼,,𝑛   𝑅,𝑓,𝑛   𝐷,𝑔   𝐵,𝑓
Allowed substitution hints:   𝜑(𝑓,𝑔,,𝑛)   𝐵(𝑔,,𝑛)   𝐷(𝑓,,𝑛)   𝑃(𝑓,𝑔,,𝑛)   𝑅(𝑔,)   𝐻(𝑓,𝑔,,𝑛)   𝐼(𝑔)   𝑉(𝑓,𝑔,,𝑛)   𝑊(𝑓,𝑔,,𝑛)   0 (𝑓,𝑔,,𝑛)

Proof of Theorem mhpfval
Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhpfval.h . 2 𝐻 = (𝐼 mHomP 𝑅)
2 mhpfval.i . . . 4 (𝜑𝐼𝑉)
32elexd 3489 . . 3 (𝜑𝐼 ∈ V)
4 mhpfval.r . . . 4 (𝜑𝑅𝑊)
54elexd 3489 . . 3 (𝜑𝑅 ∈ V)
6 oveq12 7149 . . . . . . . . 9 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
7 mhpfval.p . . . . . . . . 9 𝑃 = (𝐼 mPoly 𝑅)
86, 7eqtr4di 2875 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
98fveq2d 6656 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
10 mhpfval.b . . . . . . 7 𝐵 = (Base‘𝑃)
119, 10eqtr4di 2875 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
12 fveq2 6652 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 mhpfval.0 . . . . . . . . . 10 0 = (0g𝑅)
1412, 13eqtr4di 2875 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1514oveq2d 7156 . . . . . . . 8 (𝑟 = 𝑅 → (𝑓 supp (0g𝑟)) = (𝑓 supp 0 ))
1615adantl 485 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑓 supp (0g𝑟)) = (𝑓 supp 0 ))
17 oveq2 7148 . . . . . . . . . . 11 (𝑖 = 𝐼 → (ℕ0m 𝑖) = (ℕ0m 𝐼))
1817rabeqdv 3460 . . . . . . . . . 10 (𝑖 = 𝐼 → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
19 mhpfval.d . . . . . . . . . 10 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
2018, 19eqtr4di 2875 . . . . . . . . 9 (𝑖 = 𝐼 → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
2120rabeqdv 3460 . . . . . . . 8 (𝑖 = 𝐼 → {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛} = {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛})
2221adantr 484 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛} = {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛})
2316, 22sseq12d 3975 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → ((𝑓 supp (0g𝑟)) ⊆ {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛} ↔ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}))
2411, 23rabeqbidv 3461 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g𝑟)) ⊆ {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛}} = {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}})
2524mpteq2dv 5138 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g𝑟)) ⊆ {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}) = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
26 df-mhp 20783 . . . 4 mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g𝑟)) ⊆ {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
27 nn0ex 11891 . . . . 5 0 ∈ V
2827mptex 6968 . . . 4 (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}) ∈ V
2925, 26, 28ovmpoa 7289 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mHomP 𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
303, 5, 29syl2anc 587 . 2 (𝜑 → (𝐼 mHomP 𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
311, 30syl5eq 2869 1 (𝜑𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  {crab 3134  Vcvv 3469   ⊆ wss 3908   ↦ cmpt 5122  ◡ccnv 5531   “ cima 5535  ‘cfv 6334  (class class class)co 7140   supp csupp 7817   ↑m cmap 8393  Fincfn 8496  ℕcn 11625  ℕ0cn0 11885  Basecbs 16474   ↾s cress 16475  0gc0g 16704   Σg cgsu 16705  ℂfldccnfld 20089   mPoly cmpl 20589   mHomP cmhp 20779 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-1cn 10584  ax-addcl 10586 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-nn 11626  df-n0 11886  df-mhp 20783 This theorem is referenced by:  mhpval  20790
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