Theorem pserval 26366

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 7433	. . . 4 ⊢ (𝑦 = 𝑋 → (𝑦↑𝑚) = (𝑋↑𝑚))
2	1	oveq2d 7442	. . 3 ⊢ (𝑦 = 𝑋 → ((𝐴‘𝑚) · (𝑦↑𝑚)) = ((𝐴‘𝑚) · (𝑋↑𝑚)))
3	2	mpteq2dv 5254	. 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 6902	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝐴‘𝑛) = (𝐴‘𝑚))
6		oveq2 7434	. . . . . . 7 ⊢ (𝑛 = 𝑚 → (𝑥↑𝑛) = (𝑥↑𝑚))
7	5, 6	oveq12d 7444	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝐴‘𝑛) · (𝑥↑𝑛)) = ((𝐴‘𝑚) · (𝑥↑𝑚)))
8	7	cbvmptv 5265	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑥↑𝑚)))
9		oveq1 7433	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥↑𝑚) = (𝑦↑𝑚))
10	9	oveq2d 7442	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴‘𝑚) · (𝑥↑𝑚)) = ((𝐴‘𝑚) · (𝑦↑𝑚)))
11	10	mpteq2dv 5254	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑥↑𝑚))) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑦↑𝑚))))
12	8, 11	eqtrid 2780	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛))) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑦↑𝑚))))
13	12	cbvmptv 5265	. . 3 ⊢ (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) = (𝑦 ∈ ℂ ↦ (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑦↑𝑚))))
14	4, 13	eqtri 2756	. 2 ⊢ 𝐺 = (𝑦 ∈ ℂ ↦ (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑦↑𝑚))))
15		nn0ex 12516	. . 3 ⊢ ℕ0 ∈ V
16	15	mptex 7241	. 2 ⊢ (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑋↑𝑚))) ∈ V
17	3, 14, 16	fvmpt 7010	1 ⊢ (𝑋 ∈ ℂ → (𝐺‘𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑋↑𝑚))))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5235  ‘cfv 6553  (class class class)co 7426  ℂcc 11144   · cmul 11151  ℕ₀cn0 12510  ↑cexp 14066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-1cn 11204  ax-addcl 11206

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-nn 12251  df-n0 12511

This theorem is referenced by:  pserval2  26367  psergf  26368