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Theorem mhpval 21530
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 21529 . 2 (𝜑𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}}))
9 eqeq2 2748 . . . . . 6 (𝑛 = 𝑁 → (((ℂflds0) Σg 𝑔) = 𝑛 ↔ ((ℂflds0) Σg 𝑔) = 𝑁))
109rabbidv 3415 . . . . 5 (𝑛 = 𝑁 → {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛} = {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁})
1110sseq2d 3976 . . . 4 (𝑛 = 𝑁 → ((𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛} ↔ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}))
1211rabbidv 3415 . . 3 (𝑛 = 𝑁 → {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑛}} = {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}})
1312adantl 482 . 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 6856 . . . 4 𝐵 ∈ V
1615rabex 5289 . . 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 6955 1 (𝜑 → (𝐻𝑁) = {𝑓𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {crab 3407  Vcvv 3445  wss 3910  ccnv 5632  cima 5636  cfv 6496  (class class class)co 7357   supp csupp 8092  m cmap 8765  Fincfn 8883  cn 12153  0cn0 12413  Basecbs 17083  s cress 17112  0gc0g 17321   Σg cgsu 17322  fldccnfld 20796   mPoly cmpl 21308   mHomP cmhp 21519
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-nn 12154  df-n0 12414  df-mhp 21523
This theorem is referenced by:  ismhp  21531
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