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Mirrors > Home > MPE Home > Th. List > pserval2 | Structured version Visualization version GIF version |
Description: Value of the function 𝐺 that gives the sequence of monomials of a power series. (Contributed by Mario Carneiro, 26-Feb-2015.) |
Ref | Expression |
---|---|
pser.g | ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) |
Ref | Expression |
---|---|
pserval2 | ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐺‘𝑋)‘𝑁) = ((𝐴‘𝑁) · (𝑋↑𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pser.g | . . . 4 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐴‘𝑛) · (𝑥↑𝑛)))) | |
2 | 1 | pserval 25569 | . . 3 ⊢ (𝑋 ∈ ℂ → (𝐺‘𝑋) = (𝑦 ∈ ℕ0 ↦ ((𝐴‘𝑦) · (𝑋↑𝑦)))) |
3 | 2 | fveq1d 6776 | . 2 ⊢ (𝑋 ∈ ℂ → ((𝐺‘𝑋)‘𝑁) = ((𝑦 ∈ ℕ0 ↦ ((𝐴‘𝑦) · (𝑋↑𝑦)))‘𝑁)) |
4 | fveq2 6774 | . . . 4 ⊢ (𝑦 = 𝑁 → (𝐴‘𝑦) = (𝐴‘𝑁)) | |
5 | oveq2 7283 | . . . 4 ⊢ (𝑦 = 𝑁 → (𝑋↑𝑦) = (𝑋↑𝑁)) | |
6 | 4, 5 | oveq12d 7293 | . . 3 ⊢ (𝑦 = 𝑁 → ((𝐴‘𝑦) · (𝑋↑𝑦)) = ((𝐴‘𝑁) · (𝑋↑𝑁))) |
7 | eqid 2738 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ ((𝐴‘𝑦) · (𝑋↑𝑦))) = (𝑦 ∈ ℕ0 ↦ ((𝐴‘𝑦) · (𝑋↑𝑦))) | |
8 | ovex 7308 | . . 3 ⊢ ((𝐴‘𝑁) · (𝑋↑𝑁)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6875 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑦 ∈ ℕ0 ↦ ((𝐴‘𝑦) · (𝑋↑𝑦)))‘𝑁) = ((𝐴‘𝑁) · (𝑋↑𝑁))) |
10 | 3, 9 | sylan9eq 2798 | 1 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐺‘𝑋)‘𝑁) = ((𝐴‘𝑁) · (𝑋↑𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 · cmul 10876 ℕ0cn0 12233 ↑cexp 13782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-1cn 10929 ax-addcl 10931 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-nn 11974 df-n0 12234 |
This theorem is referenced by: radcnvlem1 25572 radcnv0 25575 dvradcnv 25580 pserulm 25581 psercn2 25582 pserdvlem2 25587 abelth 25600 |
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