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Theorem mhpval 20796
 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)
mhpval.n (𝜑𝑁 ∈ ℕ0)
Assertion
Ref Expression
mhpval (𝜑 → (𝐻𝑁) = {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}})
Distinct variable groups:   𝑓,𝑔,   𝑓,𝐼,   𝑅,𝑓   𝐷,𝑔   𝐵,𝑓   𝑓,𝑁,𝑔
Allowed substitution hints:   𝜑(𝑓,𝑔,)   𝐵(𝑔,)   𝐷(𝑓,)   𝑃(𝑓,𝑔,)   𝑅(𝑔,)   𝐻(𝑓,𝑔,)   𝐼(𝑔)   𝑁()   𝑉(𝑓,𝑔,)   𝑊(𝑓,𝑔,)   0 (𝑓,𝑔,)

Proof of Theorem mhpval
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 mhpfval.h . . 3 𝐻 = (𝐼 mHomP 𝑅)
2 mhpfval.p . . 3 𝑃 = (𝐼 mPoly 𝑅)
3 mhpfval.b . . 3 𝐵 = (Base‘𝑃)
4 mhpfval.0 . . 3 0 = (0g𝑅)
5 mhpfval.d . . 3 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
6 mhpfval.i . . 3 (𝜑𝐼𝑉)
7 mhpfval.r . . 3 (𝜑𝑅𝑊)
81, 2, 3, 4, 5, 6, 7mhpfval 20795 . 2 (𝜑𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
9 eqeq2 2813 . . . . . 6 (𝑛 = 𝑁 → (((ℂflds0) Σg 𝑔) = 𝑛 ↔ ((ℂflds0) Σg 𝑔) = 𝑁))
109rabbidv 3430 . . . . 5 (𝑛 = 𝑁 → {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛} = {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁})
1110sseq2d 3950 . . . 4 (𝑛 = 𝑁 → ((𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛} ↔ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}))
1211rabbidv 3430 . . 3 (𝑛 = 𝑁 → {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}} = {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}})
1312adantl 485 . 2 ((𝜑𝑛 = 𝑁) → {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}} = {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}})
14 mhpval.n . 2 (𝜑𝑁 ∈ ℕ0)
153fvexi 6663 . . . 4 𝐵 ∈ V
1615rabex 5202 . . 3 {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}} ∈ V
1716a1i 11 . 2 (𝜑 → {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}} ∈ V)
188, 13, 14, 17fvmptd 6756 1 (𝜑 → (𝐻𝑁) = {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  {crab 3113  Vcvv 3444   ⊆ wss 3884  ◡ccnv 5522   “ cima 5526  ‘cfv 6328  (class class class)co 7139   supp csupp 7817   ↑m cmap 8393  Fincfn 8496  ℕcn 11629  ℕ0cn0 11889  Basecbs 16479   ↾s cress 16480  0gc0g 16709   Σg cgsu 16710  ℂfldccnfld 20095   mPoly cmpl 20595   mHomP cmhp 20785 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-1cn 10588  ax-addcl 10590 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-nn 11630  df-n0 11890  df-mhp 20789 This theorem is referenced by:  ismhp  20797
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