Theorem mhpfval 20327

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 = (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 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}}))

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

Allowed substitution hints:   𝜑(𝑓,𝑔,ℎ,𝑗,𝑛)   𝐵(𝑔,ℎ,𝑗,𝑛)   𝐷(𝑓,ℎ,𝑗,𝑛)   𝑃(𝑓,𝑔,ℎ,𝑗,𝑛)   𝑅(𝑔,ℎ,𝑗)   𝐻(𝑓,𝑔,ℎ,𝑗,𝑛)   𝐼(𝑔,𝑗)   𝑉(𝑓,𝑔,ℎ,𝑗,𝑛)   𝑊(𝑓,𝑔,ℎ,𝑗,𝑛)   0 (𝑓,𝑔,ℎ,𝑗,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 3511	. . 3 ⊢ (𝜑 → 𝐼 ∈ V)
4		mhpfval.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
5	4	elexd 3511	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
6		nn0ex 11897	. . . . 5 ⊢ ℕ0 ∈ V
7	6	mptex 6979	. . . 4 ⊢ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}}) ∈ V
8	7	a1i 11	. . 3 ⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}}) ∈ V)
9		oveq12 7158	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = (𝐼 mPoly 𝑅))
10		mhpfval.p	. . . . . . . . 9 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
11	9, 10	syl6eqr 2873	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPoly 𝑟) = 𝑃)
12	11	fveq2d 6667	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = (Base‘𝑃))
13		mhpfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑃)
14	12, 13	syl6eqr 2873	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPoly 𝑟)) = 𝐵)
15		fveq2 6663	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
16		mhpfval.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑅)
17	15, 16	syl6eqr 2873	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
18	17	oveq2d 7165	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑓 supp (0g‘𝑟)) = (𝑓 supp 0 ))
19	18	adantl 484	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓 supp (0g‘𝑟)) = (𝑓 supp 0 ))
20		oveq2 7157	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
21	20	rabeqdv 3481	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
22		mhpfval.d	. . . . . . . . . 10 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
23	21, 22	syl6eqr 2873	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
24	23	rabeqdv 3481	. . . . . . . 8 ⊢ (𝑖 = 𝐼 → {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛} = {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛})
25	24	adantr 483	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛} = {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛})
26	19, 25	sseq12d 3993	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛} ↔ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}))
27	14, 26	rabeqbidv 3482	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}} = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}})
28	27	mpteq2dv 5155	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ∣ (𝑓 supp (0g‘𝑟)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}}) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}}))
29		df-mhp 20321	. . . 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} ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}}))
30	28, 29	ovmpoga 7297	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V ∧ (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}}) ∈ V) → (𝐼 mHomP 𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}}))
31	3, 5, 8, 30	syl3anc 1366	. 2 ⊢ (𝜑 → (𝐼 mHomP 𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}}))
32	1, 31	syl5eq 2867	1 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}}))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  {crab 3141  Vcvv 3491   ⊆ wss 3929   ↦ cmpt 5139  ^◡ccnv 5547   “ cima 5551  ‘cfv 6348  (class class class)co 7149   supp csupp 7823   ↑_m cmap 8399  Fincfn 8502  ℕcn 11631  ℕ₀cn0 11891  Σcsu 15037  Basecbs 16478  0_gc0g 16708   mPoly cmpl 20128   mHomP cmhp 20317

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454  ax-cnex 10586  ax-1cn 10588  ax-addcl 10590

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3493  df-sbc 3769  df-csb 3877  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7574  df-wrecs 7940  df-recs 8001  df-rdg 8039  df-nn 11632  df-n0 11892  df-mhp 20321

This theorem is referenced by:  mhpval  20328