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Theorem mhpfval 21327
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 3451 . . 3 (𝜑𝐼 ∈ V)
4 mhpfval.r . . . 4 (𝜑𝑅𝑊)
54elexd 3451 . . 3 (𝜑𝑅 ∈ V)
6 oveq12 7286 . . . . . . . . 9 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
7 mhpfval.p . . . . . . . . 9 𝑃 = (𝐼 mPoly 𝑅)
86, 7eqtr4di 2796 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
98fveq2d 6780 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
10 mhpfval.b . . . . . . 7 𝐵 = (Base‘𝑃)
119, 10eqtr4di 2796 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
12 fveq2 6776 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 mhpfval.0 . . . . . . . . . 10 0 = (0g𝑅)
1412, 13eqtr4di 2796 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1514oveq2d 7293 . . . . . . . 8 (𝑟 = 𝑅 → (𝑓 supp (0g𝑟)) = (𝑓 supp 0 ))
1615adantl 482 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑓 supp (0g𝑟)) = (𝑓 supp 0 ))
17 oveq2 7285 . . . . . . . . . . 11 (𝑖 = 𝐼 → (ℕ0m 𝑖) = (ℕ0m 𝐼))
1817rabeqdv 3418 . . . . . . . . . 10 (𝑖 = 𝐼 → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
19 mhpfval.d . . . . . . . . . 10 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
2018, 19eqtr4di 2796 . . . . . . . . 9 (𝑖 = 𝐼 → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
2120rabeqdv 3418 . . . . . . . 8 (𝑖 = 𝐼 → {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛} = {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛})
2221adantr 481 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛} = {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛})
2316, 22sseq12d 3955 . . . . . 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 3419 . . . . 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 5178 . . . 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 21321 . . . 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 12237 . . . . 5 0 ∈ V
2827mptex 7101 . . . 4 (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}) ∈ V
2925, 26, 28ovmpoa 7428 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mHomP 𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
303, 5, 29syl2anc 584 . 2 (𝜑 → (𝐼 mHomP 𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
311, 30eqtrid 2790 1 (𝜑𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  {crab 3068  Vcvv 3431  wss 3888  cmpt 5159  ccnv 5590  cima 5594  cfv 6435  (class class class)co 7277   supp csupp 7975  m cmap 8613  Fincfn 8731  cn 11971  0cn0 12231  Basecbs 16910  s cress 16939  0gc0g 17148   Σg cgsu 17149  fldccnfld 20595   mPoly cmpl 21107   mHomP cmhp 21317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5211  ax-sep 5225  ax-nul 5232  ax-pr 5354  ax-un 7588  ax-cnex 10925  ax-1cn 10927  ax-addcl 10929
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-tr 5194  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-pred 6204  df-ord 6271  df-on 6272  df-lim 6273  df-suc 6274  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-ov 7280  df-oprab 7281  df-mpo 7282  df-om 7713  df-2nd 7832  df-frecs 8095  df-wrecs 8126  df-recs 8200  df-rdg 8239  df-nn 11972  df-n0 12232  df-mhp 21321
This theorem is referenced by:  mhpval  21328  prjcrv0  40467
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