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Theorem psdcoef 22083
Description: Coefficient of a term of the derivative of a power series. (Contributed by SN, 12-Apr-2025.)
Hypotheses
Ref Expression
psdffval.s 𝑆 = (𝐼 mPwSer 𝑅)
psdffval.b 𝐵 = (Base‘𝑆)
psdffval.d 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
psdffval.i (𝜑𝐼𝑉)
psdffval.r (𝜑𝑅𝑊)
psdfval.x (𝜑𝑋𝐼)
psdval.f (𝜑𝐹𝐵)
psdcoef.k (𝜑𝐾𝐷)
Assertion
Ref Expression
psdcoef (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝐾) = (((𝐾𝑋) + 1)(.g𝑅)(𝐹‘(𝐾f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
Distinct variable groups:   ,𝐼,𝑦   𝑦,𝑋
Allowed substitution hints:   𝜑(𝑦,)   𝐵(𝑦,)   𝐷(𝑦,)   𝑅(𝑦,)   𝑆(𝑦,)   𝐹(𝑦,)   𝐾(𝑦,)   𝑉(𝑦,)   𝑊(𝑦,)   𝑋()

Proof of Theorem psdcoef
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 psdffval.s . . 3 𝑆 = (𝐼 mPwSer 𝑅)
2 psdffval.b . . 3 𝐵 = (Base‘𝑆)
3 psdffval.d . . 3 𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
4 psdffval.i . . 3 (𝜑𝐼𝑉)
5 psdffval.r . . 3 (𝜑𝑅𝑊)
6 psdfval.x . . 3 (𝜑𝑋𝐼)
7 psdval.f . . 3 (𝜑𝐹𝐵)
81, 2, 3, 4, 5, 6, 7psdval 22082 . 2 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) = (𝑘𝐷 ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
9 fveq1 6896 . . . . 5 (𝑘 = 𝐾 → (𝑘𝑋) = (𝐾𝑋))
109oveq1d 7435 . . . 4 (𝑘 = 𝐾 → ((𝑘𝑋) + 1) = ((𝐾𝑋) + 1))
11 fvoveq1 7443 . . . 4 (𝑘 = 𝐾 → (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝐹‘(𝐾f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
1210, 11oveq12d 7438 . . 3 (𝑘 = 𝐾 → (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = (((𝐾𝑋) + 1)(.g𝑅)(𝐹‘(𝐾f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
1312adantl 481 . 2 ((𝜑𝑘 = 𝐾) → (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) = (((𝐾𝑋) + 1)(.g𝑅)(𝐹‘(𝐾f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
14 psdcoef.k . 2 (𝜑𝐾𝐷)
15 ovexd 7455 . 2 (𝜑 → (((𝐾𝑋) + 1)(.g𝑅)(𝐹‘(𝐾f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) ∈ V)
168, 13, 14, 15fvmptd 7012 1 (𝜑 → ((((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹)‘𝐾) = (((𝐾𝑋) + 1)(.g𝑅)(𝐹‘(𝐾f + (𝑦𝐼 ↦ 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  cn 12242  0cn0 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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