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

Theorem pserval 23968
 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 6556 . . . 4 (𝑦 = 𝑋 → (𝑦𝑚) = (𝑋𝑚))
21oveq2d 6565 . . 3 (𝑦 = 𝑋 → ((𝐴𝑚) · (𝑦𝑚)) = ((𝐴𝑚) · (𝑋𝑚)))
32mpteq2dv 4673 . 2 (𝑦 = 𝑋 → (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))))
4 pser.g . . 3 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))))
5 fveq2 6103 . . . . . . 7 (𝑛 = 𝑚 → (𝐴𝑛) = (𝐴𝑚))
6 oveq2 6557 . . . . . . 7 (𝑛 = 𝑚 → (𝑥𝑛) = (𝑥𝑚))
75, 6oveq12d 6567 . . . . . 6 (𝑛 = 𝑚 → ((𝐴𝑛) · (𝑥𝑛)) = ((𝐴𝑚) · (𝑥𝑚)))
87cbvmptv 4678 . . . . 5 (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑥𝑚)))
9 oveq1 6556 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑚) = (𝑦𝑚))
109oveq2d 6565 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑚) · (𝑥𝑚)) = ((𝐴𝑚) · (𝑦𝑚)))
1110mpteq2dv 4673 . . . . 5 (𝑥 = 𝑦 → (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑥𝑚))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
128, 11syl5eq 2656 . . . 4 (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛))) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
1312cbvmptv 4678 . . 3 (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) · (𝑥𝑛)))) = (𝑦 ∈ ℂ ↦ (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
144, 13eqtri 2632 . 2 𝐺 = (𝑦 ∈ ℂ ↦ (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑦𝑚))))
15 nn0ex 11175 . . 3 0 ∈ V
1615mptex 6390 . 2 (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))) ∈ V
173, 14, 16fvmpt 6191 1 (𝑋 ∈ ℂ → (𝐺𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴𝑚) · (𝑋𝑚))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   · cmul 9820  ℕ0cn0 11169  ↑cexp 12722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-n0 11170 This theorem is referenced by:  pserval2  23969  psergf  23970
 Copyright terms: Public domain W3C validator