Theorem psdffval 22100

Description: Value of the power series differentiation operation. (Contributed by SN, 11-Apr-2025.)

Hypotheses

Ref	Expression
psdffval.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psdffval.b	⊢ 𝐵 = (Base‘𝑆)
psdffval.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
psdffval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psdffval.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)

Assertion

Ref	Expression
psdffval	⊢ (𝜑 → (𝐼 mPSDer 𝑅) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑥) + 1)(.g‘𝑅)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))))))))

Distinct variable groups:   𝑓,𝐼,ℎ,𝑘,𝑥,𝑦   𝑅,𝑓,𝑘,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,ℎ,𝑘)   𝐵(𝑥,𝑦,𝑓,ℎ,𝑘)   𝐷(𝑥,𝑦,𝑓,ℎ,𝑘)   𝑅(𝑦,ℎ)   𝑆(𝑥,𝑦,𝑓,ℎ,𝑘)   𝑉(𝑥,𝑦,𝑓,ℎ,𝑘)   𝑊(𝑥,𝑦,𝑓,ℎ,𝑘)

Proof of Theorem psdffval

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

Step	Hyp	Ref	Expression
1		df-psd 22099	. . 3 ⊢ mPSDer = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑥) + 1)(.g‘𝑟)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)))))))))
2	1	a1i 11	. 2 ⊢ (𝜑 → mPSDer = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑥) + 1)(.g‘𝑟)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0))))))))))
3		simpl 482	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑖 = 𝐼)
4		oveq12 7419	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
5		psdffval.s	. . . . . . . 8 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
6	4, 5	eqtr4di 2789	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = 𝑆)
7	6	fveq2d 6885	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPwSer 𝑟)) = (Base‘𝑆))
8		psdffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
9	7, 8	eqtr4di 2789	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPwSer 𝑟)) = 𝐵)
10	3	oveq2d 7426	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
11	10	rabeqdv 3436	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin})
12		psdffval.d	. . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
13	11, 12	eqtr4di 2789	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
14		fveq2 6881	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (.g‘𝑟) = (.g‘𝑅))
15	14	adantl 481	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (.g‘𝑟) = (.g‘𝑅))
16		eqidd 2737	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑘‘𝑥) + 1) = ((𝑘‘𝑥) + 1))
17	3	mpteq1d 5215	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
18	17	oveq2d 7426	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0))) = (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))
19	18	fveq2d 6885	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)))) = (𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))))
20	15, 16, 19	oveq123d 7431	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (((𝑘‘𝑥) + 1)(.g‘𝑟)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0))))) = (((𝑘‘𝑥) + 1)(.g‘𝑅)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))))
21	13, 20	mpteq12dv 5212	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑥) + 1)(.g‘𝑟)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)))))) = (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑥) + 1)(.g‘𝑅)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))))))
22	9, 21	mpteq12dv 5212	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑥) + 1)(.g‘𝑟)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0))))))) = (𝑓 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑥) + 1)(.g‘𝑅)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))))))
23	3, 22	mpteq12dv 5212	. . 3 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑥) + 1)(.g‘𝑟)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)))))))) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑥) + 1)(.g‘𝑅)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))))))))
24	23	adantl 481	. 2 ⊢ ((𝜑 ∧ (𝑖 = 𝐼 ∧ 𝑟 = 𝑅)) → (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘(𝑖 mPwSer 𝑟)) ↦ (𝑘 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (((𝑘‘𝑥) + 1)(.g‘𝑟)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)))))))) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑥) + 1)(.g‘𝑅)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))))))))
25		psdffval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
26	25	elexd 3488	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
27		psdffval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
28	27	elexd 3488	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
29	25	mptexd 7221	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑥) + 1)(.g‘𝑅)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))))))) ∈ V)
30	2, 24, 26, 28, 29	ovmpod 7564	1 ⊢ (𝜑 → (𝐼 mPSDer 𝑅) = (𝑥 ∈ 𝐼 ↦ (𝑓 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑥) + 1)(.g‘𝑅)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3420  Vcvv 3464  ifcif 4505   ↦ cmpt 5206  ^◡ccnv 5658   “ cima 5662  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ∘_f cof 7674   ↑_m cmap 8845  Fincfn 8964  0cc0 11134  1c1 11135   + caddc 11137  ℕcn 12245  ℕ₀cn0 12506  Basecbs 17233  ._gcmg 19055   mPwSer cmps 21869   mPSDer cpsd 22073

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-psd 22099

This theorem is referenced by:  psdfval  22101