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Theorem psdcoef 22032
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 22031 . 2 (𝜑 → (((𝐼 mPSDer 𝑅)‘𝑋)‘𝐹) = (𝑘𝐷 ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
9 fveq1 6881 . . . . 5 (𝑘 = 𝐾 → (𝑘𝑋) = (𝐾𝑋))
109oveq1d 7417 . . . 4 (𝑘 = 𝐾 → ((𝑘𝑋) + 1) = ((𝐾𝑋) + 1))
11 fvoveq1 7425 . . . 4 (𝑘 = 𝐾 → (𝐹‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) = (𝐹‘(𝐾f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
1210, 11oveq12d 7420 . . 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 7437 . 2 (𝜑 → (((𝐾𝑋) + 1)(.g𝑅)(𝐹‘(𝐾f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))) ∈ V)
168, 13, 14, 15fvmptd 6996 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 1533  wcel 2098  {crab 3424  Vcvv 3466  ifcif 4521  cmpt 5222  ccnv 5666  cima 5670  cfv 6534  (class class class)co 7402  f cof 7662  m cmap 8817  Fincfn 8936  0cc0 11107  1c1 11108   + caddc 11110  cn 12211  0cn0 12471  Basecbs 17149  .gcmg 18991   mPwSer cmps 21787   mPSDer cpsd 22004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-psd 22028
This theorem is referenced by:  psdvsca  22036
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