Theorem psdffval 22301

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 22300	. . 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 487	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → 𝑖 = 𝐼)
4		oveq12 7421	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = (𝐼 mPwSer 𝑅))
5		psdffval.s	. . . . . . . 8 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
6	4, 5	eqtr4di 2816	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 mPwSer 𝑟) = 𝑆)
7	6	fveq2d 6887	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPwSer 𝑟)) = (Base‘𝑆))
8		psdffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
9	7, 8	eqtr4di 2816	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘(𝑖 mPwSer 𝑟)) = 𝐵)
10	3	oveq2d 7428	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (ℕ0 ↑m 𝑖) = (ℕ0 ↑m 𝐼))
11	10	rabeqdv 3431	. . . . . . 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 2816	. . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} = 𝐷)
14		fveq2 6883	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (.g‘𝑟) = (.g‘𝑅))
15	14	adantl 486	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (.g‘𝑟) = (.g‘𝑅))
16		eqidd 2764	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑘‘𝑥) + 1) = ((𝑘‘𝑥) + 1))
17	3	mpteq1d 5202	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))
18	17	oveq2d 7428	. . . . . . . 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 6887	. . . . . . 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 7433	. . . . . 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 5199	. . . . 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 5199	. . . 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 5199	. . 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 486	. 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 3478	. 2 ⊢ (𝜑 → 𝐼 ∈ V)
27		psdffval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
28	27	elexd 3478	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
29	25	mptexd 7224	. 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 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455  ifcif 4488   ↦ cmpt 5193  ^◡ccnv 5662   “ cima 5666  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414   ∘_f cof 7674   ↑_m cmap 8825  Fincfn 8944  0cc0 11101  1c1 11102   + caddc 11104  ℕcn 12234  ℕ₀cn0 12505  Basecbs 17270  ._gcmg 19134   mPwSer cmps 22035   mPSDer cpsd 22278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  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-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-psd 22300

This theorem is referenced by:  psdfval  22302