Theorem mhpfval 22081

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 3464	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
4		mhpfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
5	4	elexd 3464	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
6		oveq12 7367	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
7		mhpfval.p	. . . . . . . . 9 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
8	6, 7	eqtr4di 2789	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
9	8	fveq2d 6838	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
10		mhpfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑃)
11	9, 10	eqtr4di 2789	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
12		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
13		mhpfval.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑅)
14	12, 13	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
15	14	oveq2d 7374	. . . . . . . 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 7366	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
18	17	rabeqdv 3414	. . . . . . . . . 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 2789	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
21	20	rabeqdv 3414	. . . . . . . 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 3967	. . . . . 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 3417	. . . . 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 5192	. . . 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 22079	. . . 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 12407	. . . . 5 ⊢ ℕ0 ∈ V
28	27	mptex 7169	. . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}) ∈ V
29	25, 26, 28	ovmpoa 7513	. . 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 2783	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 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ⊆ wss 3901   ↦ cmpt 5179  ^◡ccnv 5623   “ cima 5627  ‘cfv 6492  (class class class)co 7358   supp csupp 8102   ↑_m cmap 8763  Fincfn 8883  ℕcn 12145  ℕ₀cn0 12401  Basecbs 17136   ↾_s cress 17157  0_gc0g 17359   Σ_g cgsu 17360  ℂ_fldccnfld 21309   mPoly cmpl 21862   mHomP cmhp 22072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-1cn 11084  ax-addcl 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-nn 12146  df-n0 12402  df-mhp 22079

This theorem is referenced by:  mhpval  22082  mhprcl  22086