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Mirrors > Home > MPE Home > Th. List > eflegeo | Structured version Visualization version GIF version |
Description: The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.) |
Ref | Expression |
---|---|
eflegeo.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
eflegeo.2 | ⊢ (𝜑 → 0 ≤ 𝐴) |
eflegeo.3 | ⊢ (𝜑 → 𝐴 < 1) |
Ref | Expression |
---|---|
eflegeo | ⊢ (𝜑 → (exp‘𝐴) ≤ (1 / (1 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12274 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11987 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | eqid 2821 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
4 | 3 | eftval 15424 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
5 | 4 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
6 | eflegeo.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | reeftcl 15422 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) | |
8 | 6, 7 | sylan 582 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) |
9 | oveq2 7158 | . . . . 5 ⊢ (𝑛 = 𝑘 → (𝐴↑𝑛) = (𝐴↑𝑘)) | |
10 | eqid 2821 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) = (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) | |
11 | ovex 7183 | . . . . 5 ⊢ (𝐴↑𝑘) ∈ V | |
12 | 9, 10, 11 | fvmpt 6762 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))‘𝑘) = (𝐴↑𝑘)) |
13 | 12 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))‘𝑘) = (𝐴↑𝑘)) |
14 | reexpcl 13440 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ) | |
15 | 6, 14 | sylan 582 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ) |
16 | faccl 13637 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ) | |
17 | 16 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℕ) |
18 | 17 | nnred 11647 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℝ) |
19 | 6 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℝ) |
20 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
21 | eflegeo.2 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐴) | |
22 | 21 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 ≤ 𝐴) |
23 | 19, 20, 22 | expge0d 13522 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 ≤ (𝐴↑𝑘)) |
24 | 17 | nnge1d 11679 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 1 ≤ (!‘𝑘)) |
25 | 15, 18, 23, 24 | lemulge12d 11572 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ≤ ((!‘𝑘) · (𝐴↑𝑘))) |
26 | 17 | nngt0d 11680 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 < (!‘𝑘)) |
27 | ledivmul 11510 | . . . . 5 ⊢ (((𝐴↑𝑘) ∈ ℝ ∧ (𝐴↑𝑘) ∈ ℝ ∧ ((!‘𝑘) ∈ ℝ ∧ 0 < (!‘𝑘))) → (((𝐴↑𝑘) / (!‘𝑘)) ≤ (𝐴↑𝑘) ↔ (𝐴↑𝑘) ≤ ((!‘𝑘) · (𝐴↑𝑘)))) | |
28 | 15, 15, 18, 26, 27 | syl112anc 1370 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝐴↑𝑘) / (!‘𝑘)) ≤ (𝐴↑𝑘) ↔ (𝐴↑𝑘) ≤ ((!‘𝑘) · (𝐴↑𝑘)))) |
29 | 25, 28 | mpbird 259 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ≤ (𝐴↑𝑘)) |
30 | 6 | recnd 10663 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
31 | 3 | efcllem 15425 | . . . 4 ⊢ (𝐴 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) ∈ dom ⇝ ) |
32 | 30, 31 | syl 17 | . . 3 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) ∈ dom ⇝ ) |
33 | 6, 21 | absidd 14776 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) = 𝐴) |
34 | eflegeo.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 < 1) | |
35 | 33, 34 | eqbrtrd 5080 | . . . . 5 ⊢ (𝜑 → (abs‘𝐴) < 1) |
36 | 30, 35, 13 | geolim 15220 | . . . 4 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ⇝ (1 / (1 − 𝐴))) |
37 | seqex 13365 | . . . . 5 ⊢ seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ∈ V | |
38 | ovex 7183 | . . . . 5 ⊢ (1 / (1 − 𝐴)) ∈ V | |
39 | 37, 38 | breldm 5771 | . . . 4 ⊢ (seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ⇝ (1 / (1 − 𝐴)) → seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ∈ dom ⇝ ) |
40 | 36, 39 | syl 17 | . . 3 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ∈ dom ⇝ ) |
41 | 1, 2, 5, 8, 13, 15, 29, 32, 40 | isumle 15193 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 ((𝐴↑𝑘) / (!‘𝑘)) ≤ Σ𝑘 ∈ ℕ0 (𝐴↑𝑘)) |
42 | efval 15427 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴↑𝑘) / (!‘𝑘))) | |
43 | 30, 42 | syl 17 | . 2 ⊢ (𝜑 → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴↑𝑘) / (!‘𝑘))) |
44 | expcl 13441 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ) | |
45 | 30, 44 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ) |
46 | 1, 2, 13, 45, 36 | isumclim 15106 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐴↑𝑘) = (1 / (1 − 𝐴))) |
47 | 46 | eqcomd 2827 | . 2 ⊢ (𝜑 → (1 / (1 − 𝐴)) = Σ𝑘 ∈ ℕ0 (𝐴↑𝑘)) |
48 | 41, 43, 47 | 3brtr4d 5090 | 1 ⊢ (𝜑 → (exp‘𝐴) ≤ (1 / (1 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ↦ cmpt 5138 dom cdm 5549 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 − cmin 10864 / cdiv 11291 ℕcn 11632 ℕ0cn0 11891 seqcseq 13363 ↑cexp 13423 !cfa 13627 abscabs 14587 ⇝ cli 14835 Σcsu 15036 expce 15409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-fac 13628 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 |
This theorem is referenced by: birthdaylem3 25525 logdiflbnd 25566 emcllem2 25568 |
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