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Mirrors > Home > MPE Home > Th. List > psdcoef | Structured version Visualization version GIF version |
Description: Coefficient of a term of the derivative of a power series. (Contributed by SN, 12-Apr-2025.) |
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) |
psdval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
psdcoef.k | ⊢ (𝜑 → 𝐾 ∈ 𝐷) |
Ref | Expression |
---|---|
psdcoef | ⊢ (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝐾) = (((𝐾‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝐾 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) |
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 | psdfval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
7 | psdval.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | psdval 22082 | . 2 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) = (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))) |
9 | fveq1 6896 | . . . . 5 ⊢ (𝑘 = 𝐾 → (𝑘‘𝑋) = (𝐾‘𝑋)) | |
10 | 9 | oveq1d 7435 | . . . 4 ⊢ (𝑘 = 𝐾 → ((𝑘‘𝑋) + 1) = ((𝐾‘𝑋) + 1)) |
11 | fvoveq1 7443 | . . . 4 ⊢ (𝑘 = 𝐾 → (𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝐹‘(𝐾 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) | |
12 | 10, 11 | oveq12d 7438 | . . 3 ⊢ (𝑘 = 𝐾 → (((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)))))) |
13 | 12 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝑘 = 𝐾) → (((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)))))) |
14 | psdcoef.k | . 2 ⊢ (𝜑 → 𝐾 ∈ 𝐷) | |
15 | ovexd 7455 | . 2 ⊢ (𝜑 → (((𝐾‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝐾 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) ∈ V) | |
16 | 8, 13, 14, 15 | fvmptd 7012 | 1 ⊢ (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝐾) = (((𝐾‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝐾 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))) |
Colors of variables: wff setvar class |
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: psdvsca 22087 psdmul 22089 |
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