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Theorem mhpfval 22160
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 3502 . . 3 (𝜑𝐼 ∈ V)
4 mhpfval.r . . . 4 (𝜑𝑅𝑊)
54elexd 3502 . . 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 2793 . . . . . . . 8 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
98fveq2d 6911 . . . . . . 7 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
10 mhpfval.b . . . . . . 7 𝐵 = (Base‘𝑃)
119, 10eqtr4di 2793 . . . . . 6 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
12 fveq2 6907 . . . . . . . . . 10 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
13 mhpfval.0 . . . . . . . . . 10 0 = (0g𝑅)
1412, 13eqtr4di 2793 . . . . . . . . 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 3449 . . . . . . . . . 10 (𝑖 = 𝐼 → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
19 mhpfval.d . . . . . . . . . 10 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
2018, 19eqtr4di 2793 . . . . . . . . 9 (𝑖 = 𝐼 → { ∈ (ℕ0m 𝑖) ∣ ( “ ℕ) ∈ Fin} = 𝐷)
2120rabeqdv 3449 . . . . . . . 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 4029 . . . . . 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 3452 . . . . 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 5250 . . . 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 22158 . . . 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 12530 . . . . 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 2787 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 1537  wcel 2106  {crab 3433  Vcvv 3478  wss 3963  cmpt 5231  ccnv 5688  cima 5692  cfv 6563  (class class class)co 7431   supp csupp 8184  m cmap 8865  Fincfn 8984  cn 12264  0cn0 12524  Basecbs 17245  s cress 17274  0gc0g 17486   Σg cgsu 17487  fldccnfld 21382   mPoly cmpl 21944   mHomP cmhp 22151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-n0 12525  df-mhp 22158
This theorem is referenced by:  mhpval  22161  mhprcl  22165
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