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Mirrors > Home > MPE Home > Th. List > psergf | Structured version Visualization version GIF version |
Description: The sequence of terms in the infinite sequence defining a power series for fixed 𝑋. (Contributed by Mario Carneiro, 26-Feb-2015.) |
Ref | Expression |
---|---|
pser.g | ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) |
radcnv.a | ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
psergf.x | ⊢ (𝜑 → 𝑋 ∈ ℂ) |
Ref | Expression |
---|---|
psergf | ⊢ (𝜑 → (𝐺‘𝑋):ℕ0⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | radcnv.a | . 2 ⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) | |
2 | psergf.x | . 2 ⊢ (𝜑 → 𝑋 ∈ ℂ) | |
3 | pser.g | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
4 | 3 | pserval 26261 | . . . 4 ⊢ (𝑋 ∈ ℂ → (𝐺‘𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑋↑𝑚)))) |
5 | 4 | adantl 481 | . . 3 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑋 ∈ ℂ) → (𝐺‘𝑋) = (𝑚 ∈ ℕ0 ↦ ((𝐴‘𝑚) · (𝑋↑𝑚)))) |
6 | ffvelcdm 7083 | . . . . 5 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑚 ∈ ℕ0) → (𝐴‘𝑚) ∈ ℂ) | |
7 | 6 | adantlr 712 | . . . 4 ⊢ (((𝐴:ℕ0⟶ℂ ∧ 𝑋 ∈ ℂ) ∧ 𝑚 ∈ ℕ0) → (𝐴‘𝑚) ∈ ℂ) |
8 | expcl 14052 | . . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (𝑋↑𝑚) ∈ ℂ) | |
9 | 8 | adantll 711 | . . . 4 ⊢ (((𝐴:ℕ0⟶ℂ ∧ 𝑋 ∈ ℂ) ∧ 𝑚 ∈ ℕ0) → (𝑋↑𝑚) ∈ ℂ) |
10 | 7, 9 | mulcld 11241 | . . 3 ⊢ (((𝐴:ℕ0⟶ℂ ∧ 𝑋 ∈ ℂ) ∧ 𝑚 ∈ ℕ0) → ((𝐴‘𝑚) · (𝑋↑𝑚)) ∈ ℂ) |
11 | 5, 10 | fmpt3d 7117 | . 2 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑋 ∈ ℂ) → (𝐺‘𝑋):ℕ0⟶ℂ) |
12 | 1, 2, 11 | syl2anc 583 | 1 ⊢ (𝜑 → (𝐺‘𝑋):ℕ0⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℂcc 11114 · cmul 11121 ℕ0cn0 12479 ↑cexp 14034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-seq 13974 df-exp 14035 |
This theorem is referenced by: radcnvlem1 26264 radcnvlem2 26265 radcnvlem3 26266 radcnv0 26267 radcnvlt2 26270 dvradcnv 26272 pserulm 26273 pserdvlem2 26280 |
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