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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 |
---|---|
psdval.s | ⊢ 𝑆 = (𝐼 mPwSer 𝑅) |
psdval.b | ⊢ 𝐵 = (Base‘𝑆) |
psdval.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
psdval.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 | fveq1 6906 | . . . 4 ⊢ (𝑘 = 𝐾 → (𝑘‘𝑋) = (𝐾‘𝑋)) | |
2 | 1 | oveq1d 7446 | . . 3 ⊢ (𝑘 = 𝐾 → ((𝑘‘𝑋) + 1) = ((𝐾‘𝑋) + 1)) |
3 | fvoveq1 7454 | . . 3 ⊢ (𝑘 = 𝐾 → (𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝐹‘(𝐾 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) | |
4 | 2, 3 | oveq12d 7449 | . 2 ⊢ (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)))))) |
5 | psdval.s | . . 3 ⊢ 𝑆 = (𝐼 mPwSer 𝑅) | |
6 | psdval.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
7 | psdval.d | . . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
8 | psdval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
9 | psdval.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
10 | 5, 6, 7, 8, 9 | psdval 22181 | . 2 ⊢ (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) = (𝑘 ∈ 𝐷 ↦ (((𝑘‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝑘 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))) |
11 | psdcoef.k | . 2 ⊢ (𝜑 → 𝐾 ∈ 𝐷) | |
12 | ovexd 7466 | . 2 ⊢ (𝜑 → (((𝐾‘𝑋) + 1)(.g‘𝑅)(𝐹‘(𝐾 ∘f + (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) ∈ V) | |
13 | 4, 10, 11, 12 | fvmptd4 7040 | 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 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ifcif 4531 ↦ cmpt 5231 ◡ccnv 5688 “ cima 5692 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 ↑m cmap 8865 Fincfn 8984 0cc0 11153 1c1 11154 + caddc 11156 ℕcn 12264 ℕ0cn0 12524 Basecbs 17245 .gcmg 19098 mPwSer cmps 21942 mPSDer cpsd 22152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-slot 17216 df-ndx 17228 df-base 17246 df-psr 21947 df-psd 22178 |
This theorem is referenced by: psdvsca 22186 psdmul 22188 |
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