Theorem psdfval 22152

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 22151	. 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 6901	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑘‘𝑥) = (𝑘‘𝑋))
8	7	oveq1d 7439	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑘‘𝑥) + 1) = ((𝑘‘𝑋) + 1))
9		eqeq2 2738	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑋))
10	9	ifbid 4556	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0))
11	10	mpteq2dv 5255	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
12	11	oveq2d 7440	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0))) = (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
13	12	fveq2d 6905	. . . . . 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 7442	. . . . 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 5255	. . . 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 5255	. . 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 480	. 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 6915	. . . 4 ⊢ 𝐵 ∈ V
20	19	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
21	20	mptexd 7241	. 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 7016	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 1534   ∈ wcel 2099  {crab 3419  Vcvv 3462  ifcif 4533   ↦ cmpt 5236  ^◡ccnv 5681   “ cima 5685  ‘cfv 6554  (class class class)co 7424   ∘_f cof 7688   ↑_m cmap 8855  Fincfn 8974  0cc0 11158  1c1 11159   + caddc 11161  ℕcn 12264  ℕ₀cn0 12524  Basecbs 17213  ._gcmg 19061   mPwSer cmps 21901   mPSDer cpsd 22125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-psd 22150

This theorem is referenced by:  psdval  22153