Theorem pserval 26468

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 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))

Assertion

Ref	Expression
pserval	⊢ (𝑋 ∈ ℂ → (𝐺‘𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑋↑𝑚))))

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.

Step	Hyp	Ref	Expression
1		oveq1 7438	. . . 4 ⊢ (𝑦 = 𝑋 → (𝑦↑𝑚) = (𝑋↑𝑚))
2	1	oveq2d 7447	. . 3 ⊢ (𝑦 = 𝑋 → ((𝐴‘𝑚) · (𝑦↑𝑚)) = ((𝐴‘𝑚) · (𝑋↑𝑚)))
3	2	mpteq2dv 5250	. 2 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑦↑𝑚))) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑋↑𝑚))))
4		pser.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))))
5		fveq2 6907	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝐴‘𝑛) = (𝐴‘𝑚))
6		oveq2 7439	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑥↑𝑛) = (𝑥↑𝑚))
7	5, 6	oveq12d 7449	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑚) · (𝑥↑𝑚)))
8	7	cbvmptv 5261	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑥↑𝑚)))
9		oveq1 7438	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥↑𝑚) = (𝑦↑𝑚))
10	9	oveq2d 7447	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴‘𝑚) · (𝑥↑𝑚)) = ((𝐴‘𝑚) · (𝑦↑𝑚)))
11	10	mpteq2dv 5250	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑥↑𝑚))) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑦↑𝑚))))
12	8, 11	eqtrid 2787	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑦↑𝑚))))
13	12	cbvmptv 5261	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) = (𝑦 ∈ ℂ ↦ (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑦↑𝑚))))
14	4, 13	eqtri 2763	. 2 ⊢ 𝐺 = (𝑦 ∈ ℂ ↦ (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑦↑𝑚))))
15		nn0ex 12530	. . 3 ⊢ ℕ0 ∈ V
16	15	mptex 7243	. 2 ⊢ (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑋↑𝑚))) ∈ V
17	3, 14, 16	fvmpt 7016	1 ⊢ (𝑋 ∈ ℂ → (𝐺‘𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑋↑𝑚))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   ↦ cmpt 5231  ‘cfv 6563  (class class class)co 7431  ℂcc 11151   · cmul 11158  ℕ₀cn0 12524  ↑cexp 14099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-n0 12525

This theorem is referenced by:  pserval2  26469  psergf  26470