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Theorem mhpfval 22142
Description: Value of the "homogeneous polynomial" operator. (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 3504 . . 3 (𝜑𝐼 ∈ V)
4 mhpfval.r . . . 4 (𝜑𝑅𝑊)
54elexd 3504 . . 3 (𝜑𝑅 ∈ V)
6 oveq12 7440 . . . . . . . . 9 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
7 mhpfval.p . . . . . . . . 9 𝑃 = (𝐼 mPoly 𝑅)
86, 7eqtr4di 2795 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
98fveq2d 6910 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
10 mhpfval.b . . . . . . 7 𝐵 = (Base‘𝑃)
119, 10eqtr4di 2795 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
12 fveq2 6906 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 mhpfval.0 . . . . . . . . . 10 0 = (0g𝑅)
1412, 13eqtr4di 2795 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = 0 )
1514oveq2d 7447 . . . . . . . 8 (𝑟 = 𝑅 → (𝑓 supp (0g𝑟)) = (𝑓 supp 0 ))
1615adantl 481 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑓 supp (0g𝑟)) = (𝑓 supp 0 ))
17 oveq2 7439 . . . . . . . . . . 11 (𝑖 = 𝐼 → (ℕ0m 𝑖) = (ℕ0m 𝐼))
1817rabeqdv 3452 . . . . . . . . . 10 (𝑖 = 𝐼 → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
19 mhpfval.d . . . . . . . . . 10 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
2018, 19eqtr4di 2795 . . . . . . . . 9 (𝑖 = 𝐼 → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
2120rabeqdv 3452 . . . . . . . 8 (𝑖 = 𝐼 → {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛} = {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛})
2221adantr 480 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → {𝑔 ∈ { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑔) = 𝑛} = {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛})
2316, 22sseq12d 4017 . . . . . 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 3455 . . . . 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 5244 . . . 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 22140 . . . 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 12532 . . . . 5 0 ∈ V
2827mptex 7243 . . . 4 (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}) ∈ V
2925, 26, 28ovmpoa 7588 . . 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 2789 1 (𝜑𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951  cmpt 5225  ccnv 5684  cima 5688  cfv 6561  (class class class)co 7431   supp csupp 8185  m cmap 8866  Fincfn 8985  cn 12266  0cn0 12526  Basecbs 17247  s cress 17274  0gc0g 17484   Σg cgsu 17485  fldccnfld 21364   mPoly cmpl 21926   mHomP cmhp 22133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-n0 12527  df-mhp 22140
This theorem is referenced by:  mhpval  22143  mhprcl  22147
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