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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 24984 | . . 3 ⊢ (𝑋 ∈ ℂ → (𝐺‘𝑋) = (𝑦 ∈ ℕ0 ↦ ((𝐴‘𝑦) · (𝑋↑𝑦)))) |
3 | 2 | fveq1d 6658 | . 2 ⊢ (𝑋 ∈ ℂ → ((𝐺‘𝑋)‘𝑁) = ((𝑦 ∈ ℕ0 ↦ ((𝐴‘𝑦) · (𝑋↑𝑦)))‘𝑁)) |
4 | fveq2 6656 | . . . 4 ⊢ (𝑦 = 𝑁 → (𝐴‘𝑦) = (𝐴‘𝑁)) | |
5 | oveq2 7150 | . . . 4 ⊢ (𝑦 = 𝑁 → (𝑋↑𝑦) = (𝑋↑𝑁)) | |
6 | 4, 5 | oveq12d 7160 | . . 3 ⊢ (𝑦 = 𝑁 → ((𝐴‘𝑦) · (𝑋↑𝑦)) = ((𝐴‘𝑁) · (𝑋↑𝑁))) |
7 | eqid 2821 | . . 3 ⊢ (𝑦 ∈ ℕ0 ↦ ((𝐴‘𝑦) · (𝑋↑𝑦))) = (𝑦 ∈ ℕ0 ↦ ((𝐴‘𝑦) · (𝑋↑𝑦))) | |
8 | ovex 7175 | . . 3 ⊢ ((𝐴‘𝑁) · (𝑋↑𝑁)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6754 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((𝑦 ∈ ℕ0 ↦ ((𝐴‘𝑦) · (𝑋↑𝑦)))‘𝑁) = ((𝐴‘𝑁) · (𝑋↑𝑁))) |
10 | 3, 9 | sylan9eq 2876 | 1 ⊢ ((𝑋 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → ((𝐺‘𝑋)‘𝑁) = ((𝐴‘𝑁) · (𝑋↑𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5132 ‘cfv 6341 (class class class)co 7142 ℂcc 10521 · cmul 10528 ℕ0cn0 11884 ↑cexp 13419 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-1cn 10581 ax-addcl 10583 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-om 7567 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-nn 11625 df-n0 11885 |
This theorem is referenced by: radcnvlem1 24987 radcnv0 24990 dvradcnv 24995 pserulm 24996 psercn2 24997 pserdvlem2 25002 abelth 25015 |
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