Theorem psdfval 22081

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 22080	. 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 6897	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑘‘𝑥) = (𝑘‘𝑋))
8	7	oveq1d 7435	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑘‘𝑥) + 1) = ((𝑘‘𝑋) + 1))
9		eqeq2 2740	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑋))
10	9	ifbid 4552	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0))
11	10	mpteq2dv 5250	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
12	11	oveq2d 7436	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑥, 1, 0))) = (𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
13	12	fveq2d 6901	. . . . . 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 7438	. . . . 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 5250	. . . 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 5250	. . 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 6911	. . . 4 ⊢ 𝐵 ∈ V
20	19	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
21	20	mptexd 7236	. 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 7012	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 3429  Vcvv 3471  ifcif 4529   ↦ cmpt 5231  ^◡ccnv 5677   “ cima 5681  ‘cfv 6548  (class class class)co 7420   ∘_f cof 7683   ↑_m cmap 8844  Fincfn 8963  0cc0 11138  1c1 11139   + caddc 11141  ℕcn 12242  ℕ₀cn0 12502  Basecbs 17179  ._gcmg 19022   mPwSer cmps 21836   mPSDer cpsd 22055

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 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-psd 22079

This theorem is referenced by:  psdval  22082