Theorem mhpfval 22331

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 3480	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
4		mhpfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
5	4	elexd 3480	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
6		oveq12 7425	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
7		mhpfval.p	. . . . . . . . 9 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
8	6, 7	eqtr4di 2818	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
9	8	fveq2d 6889	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
10		mhpfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑃)
11	9, 10	eqtr4di 2818	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
12		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
13		mhpfval.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑅)
14	12, 13	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
15	14	oveq2d 7432	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑓 supp (0g‘𝑟)) = (𝑓 supp 0 ))
16	15	adantl 487	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓 supp (0g‘𝑟)) = (𝑓 supp 0 ))
17		oveq2 7424	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
18	17	rabeqdv 3433	. . . . . . . . . 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 2818	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
21	20	rabeqdv 3433	. . . . . . . 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 486	. . . . . . 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 3971	. . . . . 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 3436	. . . . 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 5207	. . . 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 22329	. . . 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 12521	. . . . 5 ⊢ ℕ0 ∈ V
28	27	mptex 7225	. . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}) ∈ V
29	25, 26, 28	ovmpoa 7571	. . 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 596	. 2 ⊢ (𝜑 → (𝐼 mHomP 𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
31	1, 30	eqtrid 2812	1 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3418  Vcvv 3457   ⊆ wss 3906   ↦ cmpt 5194  ^◡ccnv 5662   “ cima 5666  ‘cfv 6540  (class class class)co 7416   supp csupp 8158   ↑_m cmap 8826  Fincfn 8945  ℕcn 12244  ℕ₀cn0 12515  Basecbs 17287   ↾_s cress 17308  0_gc0g 17510   Σ_g cgsu 17511  ℂ_fldccnfld 21552   mPoly cmpl 22086   mHomP cmhp 22326

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-1cn 11169  ax-addcl 11171

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-nn 12245  df-n0 12516  df-mhp 22329

This theorem is used by:  mhpval  22332  mhprcl  22336