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Mirrors > Home > MPE Home > Th. List > eftlcl | Structured version Visualization version GIF version |
Description: Closure of the sum of an infinite tail of the series defining the exponential function. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
eftl.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) |
Ref | Expression |
---|---|
eftlcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2756 | . 2 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
2 | nn0z 11588 | . . 3 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ) | |
3 | 2 | adantl 473 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → 𝑀 ∈ ℤ) |
4 | eqidd 2757 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
5 | eluznn0 11946 | . . . . 5 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0) | |
6 | 5 | adantll 752 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0) |
7 | eftl.1 | . . . . 5 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
8 | 7 | eftval 15002 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
10 | simpll 807 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℂ) | |
11 | eftcl 14999 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ) | |
12 | 10, 6, 11 | syl2anc 696 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ) |
13 | 9, 12 | eqeltrd 2835 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
14 | 7 | eftlcvg 15031 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
15 | 1, 3, 4, 13, 14 | isumcl 14687 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1628 ∈ wcel 2135 ↦ cmpt 4877 ‘cfv 6045 (class class class)co 6809 ℂcc 10122 / cdiv 10872 ℕ0cn0 11480 ℤcz 11565 ℤ≥cuz 11875 ↑cexp 13050 !cfa 13250 Σcsu 14611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-rep 4919 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-inf2 8707 ax-cnex 10180 ax-resscn 10181 ax-1cn 10182 ax-icn 10183 ax-addcl 10184 ax-addrcl 10185 ax-mulcl 10186 ax-mulrcl 10187 ax-mulcom 10188 ax-addass 10189 ax-mulass 10190 ax-distr 10191 ax-i2m1 10192 ax-1ne0 10193 ax-1rid 10194 ax-rnegex 10195 ax-rrecex 10196 ax-cnre 10197 ax-pre-lttri 10198 ax-pre-lttrn 10199 ax-pre-ltadd 10200 ax-pre-mulgt0 10201 ax-pre-sup 10202 ax-addf 10203 ax-mulf 10204 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-fal 1634 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-nel 3032 df-ral 3051 df-rex 3052 df-reu 3053 df-rmo 3054 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-pss 3727 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4585 df-int 4624 df-iun 4670 df-br 4801 df-opab 4861 df-mpt 4878 df-tr 4901 df-id 5170 df-eprel 5175 df-po 5183 df-so 5184 df-fr 5221 df-se 5222 df-we 5223 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-ord 5883 df-on 5884 df-lim 5885 df-suc 5886 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-isom 6054 df-riota 6770 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-om 7227 df-1st 7329 df-2nd 7330 df-wrecs 7572 df-recs 7633 df-rdg 7671 df-1o 7725 df-oadd 7729 df-er 7907 df-pm 8022 df-en 8118 df-dom 8119 df-sdom 8120 df-fin 8121 df-sup 8509 df-inf 8510 df-oi 8576 df-card 8951 df-pnf 10264 df-mnf 10265 df-xr 10266 df-ltxr 10267 df-le 10268 df-sub 10456 df-neg 10457 df-div 10873 df-nn 11209 df-2 11267 df-3 11268 df-n0 11481 df-z 11566 df-uz 11876 df-rp 12022 df-ico 12370 df-fz 12516 df-fzo 12656 df-fl 12783 df-seq 12992 df-exp 13051 df-fac 13251 df-hash 13308 df-shft 14002 df-cj 14034 df-re 14035 df-im 14036 df-sqrt 14170 df-abs 14171 df-limsup 14397 df-clim 14414 df-rlim 14415 df-sum 14612 |
This theorem is referenced by: eftlub 15034 efsep 15035 resin4p 15063 recos4p 15064 ef01bndlem 15109 sin01bnd 15110 cos01bnd 15111 dveflem 23937 |
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