MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pserval Structured version   Visualization version   GIF version

Theorem pserval 23866
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 6432 . . . 4 (𝑦 = 𝑋 → (𝑦𝑚) = (𝑋𝑚))
21oveq2d 6441 . . 3 (𝑦 = 𝑋 → ((𝐴𝑚) · (𝑦𝑚)) = ((𝐴𝑚) · (𝑋𝑚)))
32mpteq2dv 4571 . 2 (𝑦 = 𝑋 → (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))))
4 pser.g . . 3 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))
5 fveq2 5986 . . . . . . 7 (𝑛 = 𝑚 → (𝐴𝑛) = (𝐴𝑚))
6 oveq2 6433 . . . . . . 7 (𝑛 = 𝑚 → (𝑥𝑛) = (𝑥𝑚))
75, 6oveq12d 6443 . . . . . 6 (𝑛 = 𝑚 → ((𝐴𝑛) · (𝑥𝑛)) = ((𝐴𝑚) · (𝑥𝑚)))
87cbvmptv 4576 . . . . 5 (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑥𝑚)))
9 oveq1 6432 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑚) = (𝑦𝑚))
109oveq2d 6441 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑚) · (𝑥𝑚)) = ((𝐴𝑚) · (𝑦𝑚)))
1110mpteq2dv 4571 . . . . 5 (𝑥 = 𝑦 → (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑥𝑚))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
128, 11syl5eq 2560 . . . 4 (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
1312cbvmptv 4576 . . 3 (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛)))) = (𝑦 ∈ ℂ ↦ (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
144, 13eqtri 2536 . 2 𝐺 = (𝑦 ∈ ℂ ↦ (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
15 nn0ex 11051 . . 3 0 ∈ V
1615mptex 6266 . 2 (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))) ∈ V
173, 14, 16fvmpt 6074 1 (𝑋 ∈ ℂ → (𝐺𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1938  cmpt 4541  cfv 5689  (class class class)co 6425  cc 9687   · cmul 9694  0cn0 11045  cexp 12586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-cnex 9745  ax-resscn 9746  ax-1cn 9747  ax-icn 9748  ax-addcl 9749  ax-addrcl 9750  ax-mulcl 9751  ax-mulrcl 9752  ax-i2m1 9757  ax-1ne0 9758  ax-rrecex 9761  ax-cnre 9762
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-om 6832  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-nn 10774  df-n0 11046
This theorem is referenced by:  pserval2  23867  psergf  23868
  Copyright terms: Public domain W3C validator