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Theorem pserval 26453
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
pserval (𝑋 ∈ ℂ → (𝐺𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))))
Distinct variable groups:   𝑚,𝑛,𝑥,𝐴   𝑚,𝑋   𝑚,𝐺
Allowed substitution hints:   𝐺(𝑥,𝑛)   𝑋(𝑥,𝑛)

Proof of Theorem pserval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . . 4 (𝑦 = 𝑋 → (𝑦𝑚) = (𝑋𝑚))
21oveq2d 7447 . . 3 (𝑦 = 𝑋 → ((𝐴𝑚) · (𝑦𝑚)) = ((𝐴𝑚) · (𝑋𝑚)))
32mpteq2dv 5244 . 2 (𝑦 = 𝑋 → (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))))
4 pser.g . . 3 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))
5 fveq2 6906 . . . . . . 7 (𝑛 = 𝑚 → (𝐴𝑛) = (𝐴𝑚))
6 oveq2 7439 . . . . . . 7 (𝑛 = 𝑚 → (𝑥𝑛) = (𝑥𝑚))
75, 6oveq12d 7449 . . . . . 6 (𝑛 = 𝑚 → ((𝐴𝑛) · (𝑥𝑛)) = ((𝐴𝑚) · (𝑥𝑚)))
87cbvmptv 5255 . . . . 5 (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑥𝑚)))
9 oveq1 7438 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑚) = (𝑦𝑚))
109oveq2d 7447 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑚) · (𝑥𝑚)) = ((𝐴𝑚) · (𝑦𝑚)))
1110mpteq2dv 5244 . . . . 5 (𝑥 = 𝑦 → (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑥𝑚))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
128, 11eqtrid 2789 . . . 4 (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
1312cbvmptv 5255 . . 3 (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛)))) = (𝑦 ∈ ℂ ↦ (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
144, 13eqtri 2765 . 2 𝐺 = (𝑦 ∈ ℂ ↦ (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
15 nn0ex 12532 . . 3 0 ∈ V
1615mptex 7243 . 2 (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))) ∈ V
173, 14, 16fvmpt 7016 1 (𝑋 ∈ ℂ → (𝐺𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cmpt 5225  cfv 6561  (class class class)co 7431  cc 11153   · cmul 11160  0cn0 12526  cexp 14102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-n0 12527
This theorem is referenced by:  pserval2  26454  psergf  26455
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