Theorem mhpfval 22025

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 = (0g‘𝑅)
mhpfval.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
mhpfval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mhpfval.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)

Assertion

Ref	Expression
mhpfval	⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))

Distinct variable groups:   𝑓,𝑔,ℎ,𝑛   𝑓,𝐼,ℎ,𝑛   𝑅,𝑓,𝑛   𝐷,𝑔   𝐵,𝑓

Allowed substitution hints:   𝜑(𝑓,𝑔,ℎ,𝑛)   𝐵(𝑔,ℎ,𝑛)   𝐷(𝑓,ℎ,𝑛)   𝑃(𝑓,𝑔,ℎ,𝑛)   𝑅(𝑔,ℎ)   𝐻(𝑓,𝑔,ℎ,𝑛)   𝐼(𝑔)   𝑉(𝑓,𝑔,ℎ,𝑛)   𝑊(𝑓,𝑔,ℎ,𝑛)   0 (𝑓,𝑔,ℎ,𝑛)

Proof of Theorem mhpfval

Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhpfval.h	. 2 ⊢ 𝐻 = (𝐼 mHomP 𝑅)
2		mhpfval.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
3	2	elexd 3471	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
4		mhpfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
5	4	elexd 3471	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
6		oveq12 7396	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
7		mhpfval.p	. . . . . . . . 9 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
8	6, 7	eqtr4di 2782	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
9	8	fveq2d 6862	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
10		mhpfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑃)
11	9, 10	eqtr4di 2782	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
12		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
13		mhpfval.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑅)
14	12, 13	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
15	14	oveq2d 7403	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑓 supp (0g‘𝑟)) = (𝑓 supp 0 ))
16	15	adantl 481	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓 supp (0g‘𝑟)) = (𝑓 supp 0 ))
17		oveq2 7395	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
18	17	rabeqdv 3421	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
19		mhpfval.d	. . . . . . . . . 10 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
20	18, 19	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
21	20	rabeqdv 3421	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛} = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛})
22	21	adantr 480	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛} = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛})
23	16, 22	sseq12d 3980	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛} ↔ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}))
24	11, 23	rabeqbidv 3424	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}} = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}})
25	24	mpteq2dv 5201	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
26		df-mhp 22023	. . . 4 ⊢ mHomP = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
27		nn0ex 12448	. . . . 5 ⊢ ℕ0 ∈ V
28	27	mptex 7197	. . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}) ∈ V
29	25, 26, 28	ovmpoa 7544	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mHomP 𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
30	3, 5, 29	syl2anc 584	. 2 ⊢ (𝜑 → (𝐼 mHomP 𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
31	1, 30	eqtrid 2776	1 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ⊆ wss 3914   ↦ cmpt 5188  ^◡ccnv 5637   “ cima 5641  ‘cfv 6511  (class class class)co 7387   supp csupp 8139   ↑_m cmap 8799  Fincfn 8918  ℕcn 12186  ℕ₀cn0 12442  Basecbs 17179   ↾_s cress 17200  0_gc0g 17402   Σ_g cgsu 17403  ℂ_fldccnfld 21264   mPoly cmpl 21815   mHomP cmhp 22016

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-1cn 11126  ax-addcl 11128

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-nn 12187  df-n0 12443  df-mhp 22023

This theorem is referenced by:  mhpval  22026  mhprcl  22030