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Theorem pserval2 25012
Description: Value of the function 𝐺 that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.)
Hypothesis
Ref Expression
pser.g 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))
Assertion
Ref Expression
pserval2 ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐺𝑋)‘𝑁) = ((𝐴𝑁) · (𝑋𝑁)))
Distinct variable groups:   𝑥,𝑛,𝐴   𝑛,𝑁
Allowed substitution hints:   𝐺(𝑥,𝑛)   𝑁(𝑥)   𝑋(𝑥,𝑛)

Proof of Theorem pserval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pser.g . . . 4 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))
21pserval 25011 . . 3 (𝑋 ∈ ℂ → (𝐺𝑋) = (𝑦 ∈ ℕ0 ↦ ((𝐴𝑦) · (𝑋𝑦))))
32fveq1d 6663 . 2 (𝑋 ∈ ℂ → ((𝐺𝑋)‘𝑁) = ((𝑦 ∈ ℕ0 ↦ ((𝐴𝑦) · (𝑋𝑦)))‘𝑁))
4 fveq2 6661 . . . 4 (𝑦 = 𝑁 → (𝐴𝑦) = (𝐴𝑁))
5 oveq2 7157 . . . 4 (𝑦 = 𝑁 → (𝑋𝑦) = (𝑋𝑁))
64, 5oveq12d 7167 . . 3 (𝑦 = 𝑁 → ((𝐴𝑦) · (𝑋𝑦)) = ((𝐴𝑁) · (𝑋𝑁)))
7 eqid 2824 . . 3 (𝑦 ∈ ℕ0 ↦ ((𝐴𝑦) · (𝑋𝑦))) = (𝑦 ∈ ℕ0 ↦ ((𝐴𝑦) · (𝑋𝑦)))
8 ovex 7182 . . 3 ((𝐴𝑁) · (𝑋𝑁)) ∈ V
96, 7, 8fvmpt 6759 . 2 (𝑁 ∈ ℕ0 → ((𝑦 ∈ ℕ0 ↦ ((𝐴𝑦) · (𝑋𝑦)))‘𝑁) = ((𝐴𝑁) · (𝑋𝑁)))
103, 9sylan9eq 2879 1 ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐺𝑋)‘𝑁) = ((𝐴𝑁) · (𝑋𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cmpt 5132  cfv 6343  (class class class)co 7149  cc 10533   · cmul 10540  0cn0 11894  cexp 13434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-1cn 10593  ax-addcl 10595
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-nn 11635  df-n0 11895
This theorem is referenced by:  radcnvlem1  25014  radcnv0  25017  dvradcnv  25022  pserulm  25023  psercn2  25024  pserdvlem2  25029  abelth  25042
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