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Theorem psdfval 22030
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 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
psdffval.i (𝜑𝐼𝑉)
psdffval.r (𝜑𝑅𝑊)
psdfval.x (𝜑𝑋𝐼)
Assertion
Ref Expression
psdfval (𝜑 → ((𝐼 mPSDer 𝑅)‘𝑋) = (𝑓𝐵 ↦ (𝑘𝐷 ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))))
Distinct variable groups:   𝑓,𝐼,,𝑘,𝑦   𝑅,𝑓,𝑘   𝑓,𝑋,𝑘,𝑦   𝐵,𝑓
Allowed substitution hints:   𝜑(𝑦,𝑓,,𝑘)   𝐵(𝑦,,𝑘)   𝐷(𝑦,𝑓,,𝑘)   𝑅(𝑦,)   𝑆(𝑦,𝑓,,𝑘)   𝑉(𝑦,𝑓,,𝑘)   𝑊(𝑦,𝑓,,𝑘)   𝑋()

Proof of Theorem psdfval
Dummy variable 𝑥 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)
61, 2, 3, 4, 5psdffval 22029 . 2 (𝜑 → (𝐼 mPSDer 𝑅) = (𝑥𝐼 ↦ (𝑓𝐵 ↦ (𝑘𝐷 ↦ (((𝑘𝑥) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))))))))
7 fveq2 6882 . . . . . . 7 (𝑥 = 𝑋 → (𝑘𝑥) = (𝑘𝑋))
87oveq1d 7417 . . . . . 6 (𝑥 = 𝑋 → ((𝑘𝑥) + 1) = ((𝑘𝑋) + 1))
9 eqeq2 2736 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑦 = 𝑥𝑦 = 𝑋))
109ifbid 4544 . . . . . . . . 9 (𝑥 = 𝑋 → if(𝑦 = 𝑥, 1, 0) = if(𝑦 = 𝑋, 1, 0))
1110mpteq2dv 5241 . . . . . . . 8 (𝑥 = 𝑋 → (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)) = (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
1211oveq2d 7418 . . . . . . 7 (𝑥 = 𝑋 → (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0))) = (𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
1312fveq2d 6886 . . . . . 6 (𝑥 = 𝑋 → (𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))) = (𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))
148, 13oveq12d 7420 . . . . 5 (𝑥 = 𝑋 → (((𝑘𝑥) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))) = (((𝑘𝑋) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))
1514mpteq2dv 5241 . . . 4 (𝑥 = 𝑋 → (𝑘𝐷 ↦ (((𝑘𝑥) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0)))))) = (𝑘𝐷 ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))))))
1615mpteq2dv 5241 . . 3 (𝑥 = 𝑋 → (𝑓𝐵 ↦ (𝑘𝐷 ↦ (((𝑘𝑥) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))))) = (𝑓𝐵 ↦ (𝑘𝐷 ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))))
1716adantl 481 . 2 ((𝜑𝑥 = 𝑋) → (𝑓𝐵 ↦ (𝑘𝐷 ↦ (((𝑘𝑥) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑥, 1, 0))))))) = (𝑓𝐵 ↦ (𝑘𝐷 ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))))
18 psdfval.x . 2 (𝜑𝑋𝐼)
192fvexi 6896 . . . 4 𝐵 ∈ V
2019a1i 11 . . 3 (𝜑𝐵 ∈ V)
2120mptexd 7218 . 2 (𝜑 → (𝑓𝐵 ↦ (𝑘𝐷 ↦ (((𝑘𝑋) + 1)(.g𝑅)(𝑓‘(𝑘f + (𝑦𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))))) ∈ V)
226, 17, 18, 21fvmptd 6996 1 (𝜑 → ((𝐼 mPSDer 𝑅)‘𝑋) = (𝑓𝐵 ↦ (𝑘𝐷 ↦ (((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:  psdval  22031
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