![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12797 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 12507 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | eqid 2736 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
4 | 3 | eftval 15951 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
5 | 4 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
6 | eflegeo.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | reeftcl 15949 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) | |
8 | 6, 7 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) |
9 | oveq2 7361 | . . . . 5 ⊢ (𝑛 = 𝑘 → (𝐴↑𝑛) = (𝐴↑𝑘)) | |
10 | eqid 2736 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) = (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) | |
11 | ovex 7386 | . . . . 5 ⊢ (𝐴↑𝑘) ∈ V | |
12 | 9, 10, 11 | fvmpt 6945 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))‘𝑘) = (𝐴↑𝑘)) |
13 | 12 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))‘𝑘) = (𝐴↑𝑘)) |
14 | reexpcl 13976 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ) | |
15 | 6, 14 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ) |
16 | faccl 14175 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ) | |
17 | 16 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℕ) |
18 | 17 | nnred 12164 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℝ) |
19 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℝ) |
20 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
21 | eflegeo.2 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐴) | |
22 | 21 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 ≤ 𝐴) |
23 | 19, 20, 22 | expge0d 14061 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 ≤ (𝐴↑𝑘)) |
24 | 17 | nnge1d 12197 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 1 ≤ (!‘𝑘)) |
25 | 15, 18, 23, 24 | lemulge12d 12089 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ≤ ((!‘𝑘) · (𝐴↑𝑘))) |
26 | 17 | nngt0d 12198 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 < (!‘𝑘)) |
27 | ledivmul 12027 | . . . . 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 1374 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝐴↑𝑘) / (!‘𝑘)) ≤ (𝐴↑𝑘) ↔ (𝐴↑𝑘) ≤ ((!‘𝑘) · (𝐴↑𝑘)))) |
29 | 25, 28 | mpbird 256 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ≤ (𝐴↑𝑘)) |
30 | 6 | recnd 11179 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
31 | 3 | efcllem 15952 | . . . 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 15299 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) = 𝐴) |
34 | eflegeo.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 < 1) | |
35 | 33, 34 | eqbrtrd 5125 | . . . . 5 ⊢ (𝜑 → (abs‘𝐴) < 1) |
36 | 30, 35, 13 | geolim 15747 | . . . 4 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ⇝ (1 / (1 − 𝐴))) |
37 | seqex 13900 | . . . . 5 ⊢ seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ∈ V | |
38 | ovex 7386 | . . . . 5 ⊢ (1 / (1 − 𝐴)) ∈ V | |
39 | 37, 38 | breldm 5862 | . . . 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 15721 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 ((𝐴↑𝑘) / (!‘𝑘)) ≤ Σ𝑘 ∈ ℕ0 (𝐴↑𝑘)) |
42 | efval 15954 | . . 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 13977 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ) | |
45 | 30, 44 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ) |
46 | 1, 2, 13, 45, 36 | isumclim 15634 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐴↑𝑘) = (1 / (1 − 𝐴))) |
47 | 46 | eqcomd 2742 | . 2 ⊢ (𝜑 → (1 / (1 − 𝐴)) = Σ𝑘 ∈ ℕ0 (𝐴↑𝑘)) |
48 | 41, 43, 47 | 3brtr4d 5135 | 1 ⊢ (𝜑 → (exp‘𝐴) ≤ (1 / (1 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5631 ‘cfv 6493 (class class class)co 7353 ℂcc 11045 ℝcr 11046 0cc0 11047 1c1 11048 + caddc 11050 · cmul 11052 < clt 11185 ≤ cle 11186 − cmin 11381 / cdiv 11808 ℕcn 12149 ℕ0cn0 12409 seqcseq 13898 ↑cexp 13959 !cfa 14165 abscabs 15111 ⇝ cli 15358 Σcsu 15562 expce 15936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-inf2 9573 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-pm 8764 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9374 df-inf 9375 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-n0 12410 df-z 12496 df-uz 12760 df-rp 12908 df-ico 13262 df-fz 13417 df-fzo 13560 df-fl 13689 df-seq 13899 df-exp 13960 df-fac 14166 df-hash 14223 df-shft 14944 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 df-limsup 15345 df-clim 15362 df-rlim 15363 df-sum 15563 df-ef 15942 |
This theorem is referenced by: birthdaylem3 26287 logdiflbnd 26328 emcllem2 26330 |
Copyright terms: Public domain | W3C validator |