Theorem psdfval 22101

Description: Give a map between power series and their partial derivatives with respect to a given variable 𝑋. (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)
psdfval.x	⊢ (𝜑 → 𝑋 ∈ 𝐼)

Assertion

Ref	Expression
psdfval	⊢ (𝜑 → ((𝐼 mPSDer 𝑅)‘𝑋) = (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,𝑦   𝑅,𝑓,𝑘   𝑓,𝑋,𝑘,𝑦   𝐵,𝑓

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

Proof of Theorem psdfval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psdffval.s	. . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		psdffval.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
3		psdffval.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
4		psdffval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5		psdffval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
6	1, 2, 3, 4, 5	psdffval 22100	. 2 ⊢ (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)))))))))
7		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑘‘𝑥) = (𝑘‘𝑋))
8	7	oveq1d 7425	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑘‘𝑥) + 1) = ((𝑘‘𝑋) + 1))
9		eqeq2 2748	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑋))
10	9	ifbid 4529	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0))
11	10	mpteq2dv 5220	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
12	11	oveq2d 7426	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0))) = (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
13	12	fveq2d 6885	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))) = (𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
14	8, 13	oveq12d 7428	. . . . 5 ⊢ (𝑥 = 𝑋 → (((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))))))
15	14	mpteq2dv 5220	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑘 ∈ 𝐷 ↦ (((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)))))))
16	15	mpteq2dv 5220	. . 3 ⊢ (𝑥 = 𝑋 → (𝑓 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (((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))))))))
17	16	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑓 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (((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))))))))
18		psdfval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
19	2	fvexi 6895	. . . 4 ⊢ 𝐵 ∈ V
20	19	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
21	20	mptexd 7221	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))) ∈ V)
22	6, 17, 18, 21	fvmptd 6998	1 ⊢ (𝜑 → ((𝐼 mPSDer 𝑅)‘𝑋) = (𝑓 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝑓‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3420  Vcvv 3464  ifcif 4505   ↦ cmpt 5206  ^◡ccnv 5658   “ cima 5662  ‘cfv 6536  (class class class)co 7410   ∘_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:  psdval  22102