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Mirrors > Home > MPE Home > Th. List > efcvgfsum | Structured version Visualization version GIF version |
Description: Exponential function convergence in terms of a sequence of partial finite sums. (Contributed by NM, 10-Jan-2006.) (Revised by Mario Carneiro, 28-Apr-2014.) |
Ref | Expression |
---|---|
efcvgfsum.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴↑𝑘) / (!‘𝑘))) |
Ref | Expression |
---|---|
efcvgfsum | ⊢ (𝐴 ∈ ℂ → 𝐹 ⇝ (exp‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 6914 | . . . . . . . 8 ⊢ (𝑛 = 𝑗 → (0...𝑛) = (0...𝑗)) | |
2 | 1 | sumeq1d 14809 | . . . . . . 7 ⊢ (𝑛 = 𝑗 → Σ𝑘 ∈ (0...𝑛)((𝐴↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...𝑗)((𝐴↑𝑘) / (!‘𝑘))) |
3 | efcvgfsum.1 | . . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ Σ𝑘 ∈ (0...𝑛)((𝐴↑𝑘) / (!‘𝑘))) | |
4 | sumex 14796 | . . . . . . 7 ⊢ Σ𝑘 ∈ (0...𝑗)((𝐴↑𝑘) / (!‘𝑘)) ∈ V | |
5 | 2, 3, 4 | fvmpt 6530 | . . . . . 6 ⊢ (𝑗 ∈ ℕ0 → (𝐹‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴↑𝑘) / (!‘𝑘))) |
6 | 5 | adantl 475 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = Σ𝑘 ∈ (0...𝑗)((𝐴↑𝑘) / (!‘𝑘))) |
7 | elfznn0 12728 | . . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑗) → 𝑘 ∈ ℕ0) | |
8 | 7 | adantl 475 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑗)) → 𝑘 ∈ ℕ0) |
9 | eqid 2826 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
10 | 9 | eftval 15180 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
11 | 8, 10 | syl 17 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑗)) → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
12 | simpr 479 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈ ℕ0) | |
13 | nn0uz 12005 | . . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0) | |
14 | 12, 13 | syl6eleq 2917 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈ (ℤ≥‘0)) |
15 | simpll 785 | . . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑗)) → 𝐴 ∈ ℂ) | |
16 | eftcl 15177 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ) | |
17 | 15, 8, 16 | syl2anc 581 | . . . . . 6 ⊢ (((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑗)) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ) |
18 | 11, 14, 17 | fsumser 14839 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑗)((𝐴↑𝑘) / (!‘𝑘)) = (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘𝑗)) |
19 | 6, 18 | eqtrd 2862 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘𝑗)) |
20 | 19 | ralrimiva 3176 | . . 3 ⊢ (𝐴 ∈ ℂ → ∀𝑗 ∈ ℕ0 (𝐹‘𝑗) = (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘𝑗)) |
21 | sumex 14796 | . . . . 5 ⊢ Σ𝑘 ∈ (0...𝑛)((𝐴↑𝑘) / (!‘𝑘)) ∈ V | |
22 | 21, 3 | fnmpti 6256 | . . . 4 ⊢ 𝐹 Fn ℕ0 |
23 | 0z 11716 | . . . . . 6 ⊢ 0 ∈ ℤ | |
24 | seqfn 13108 | . . . . . 6 ⊢ (0 ∈ ℤ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) Fn (ℤ≥‘0)) | |
25 | 23, 24 | ax-mp 5 | . . . . 5 ⊢ seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) Fn (ℤ≥‘0) |
26 | 13 | fneq2i 6220 | . . . . 5 ⊢ (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) Fn ℕ0 ↔ seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) Fn (ℤ≥‘0)) |
27 | 25, 26 | mpbir 223 | . . . 4 ⊢ seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) Fn ℕ0 |
28 | eqfnfv 6561 | . . . 4 ⊢ ((𝐹 Fn ℕ0 ∧ seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) Fn ℕ0) → (𝐹 = seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) ↔ ∀𝑗 ∈ ℕ0 (𝐹‘𝑗) = (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘𝑗))) | |
29 | 22, 27, 28 | mp2an 685 | . . 3 ⊢ (𝐹 = seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) ↔ ∀𝑗 ∈ ℕ0 (𝐹‘𝑗) = (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘𝑗)) |
30 | 20, 29 | sylibr 226 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐹 = seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))) |
31 | 9 | efcvg 15188 | . 2 ⊢ (𝐴 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) ⇝ (exp‘𝐴)) |
32 | 30, 31 | eqbrtrd 4896 | 1 ⊢ (𝐴 ∈ ℂ → 𝐹 ⇝ (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3118 class class class wbr 4874 ↦ cmpt 4953 Fn wfn 6119 ‘cfv 6124 (class class class)co 6906 ℂcc 10251 0cc0 10253 + caddc 10256 / cdiv 11010 ℕ0cn0 11619 ℤcz 11705 ℤ≥cuz 11969 ...cfz 12620 seqcseq 13096 ↑cexp 13155 !cfa 13354 ⇝ cli 14593 Σcsu 14794 expce 15165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-sup 8618 df-inf 8619 df-oi 8685 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-z 11706 df-uz 11970 df-rp 12114 df-ico 12470 df-fz 12621 df-fzo 12762 df-fl 12889 df-seq 13097 df-exp 13156 df-fac 13355 df-hash 13412 df-shft 14185 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-limsup 14580 df-clim 14597 df-rlim 14598 df-sum 14795 df-ef 15171 |
This theorem is referenced by: (None) |
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