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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 11598 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11266 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
3 | eqid 2610 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
4 | 3 | eftval 14646 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
5 | 4 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
6 | eflegeo.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | reeftcl 14644 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) | |
8 | 6, 7 | sylan 487 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) |
9 | oveq2 6557 | . . . . 5 ⊢ (𝑛 = 𝑘 → (𝐴↑𝑛) = (𝐴↑𝑘)) | |
10 | eqid 2610 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) = (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) | |
11 | ovex 6577 | . . . . 5 ⊢ (𝐴↑𝑘) ∈ V | |
12 | 9, 10, 11 | fvmpt 6191 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))‘𝑘) = (𝐴↑𝑘)) |
13 | 12 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))‘𝑘) = (𝐴↑𝑘)) |
14 | reexpcl 12739 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ) | |
15 | 6, 14 | sylan 487 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ) |
16 | faccl 12932 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ) | |
17 | 16 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℕ) |
18 | 17 | nnred 10912 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℝ) |
19 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℝ) |
20 | simpr 476 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
21 | eflegeo.2 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐴) | |
22 | 21 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 ≤ 𝐴) |
23 | 19, 20, 22 | expge0d 12888 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 ≤ (𝐴↑𝑘)) |
24 | 17 | nnge1d 10940 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 1 ≤ (!‘𝑘)) |
25 | 15, 18, 23, 24 | lemulge12d 10841 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ≤ ((!‘𝑘) · (𝐴↑𝑘))) |
26 | 17 | nngt0d 10941 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 < (!‘𝑘)) |
27 | ledivmul 10778 | . . . . 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 1322 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝐴↑𝑘) / (!‘𝑘)) ≤ (𝐴↑𝑘) ↔ (𝐴↑𝑘) ≤ ((!‘𝑘) · (𝐴↑𝑘)))) |
29 | 25, 28 | mpbird 246 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ≤ (𝐴↑𝑘)) |
30 | 6 | recnd 9947 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
31 | 3 | efcllem 14647 | . . . 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 14009 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) = 𝐴) |
34 | eflegeo.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 < 1) | |
35 | 33, 34 | eqbrtrd 4605 | . . . . 5 ⊢ (𝜑 → (abs‘𝐴) < 1) |
36 | 30, 35, 13 | geolim 14440 | . . . 4 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ⇝ (1 / (1 − 𝐴))) |
37 | seqex 12665 | . . . . 5 ⊢ seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ∈ V | |
38 | ovex 6577 | . . . . 5 ⊢ (1 / (1 − 𝐴)) ∈ V | |
39 | 37, 38 | breldm 5251 | . . . 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 14415 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 ((𝐴↑𝑘) / (!‘𝑘)) ≤ Σ𝑘 ∈ ℕ0 (𝐴↑𝑘)) |
42 | efval 14649 | . . 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 12740 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ) | |
45 | 30, 44 | sylan 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ) |
46 | 1, 2, 13, 45, 36 | isumclim 14330 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐴↑𝑘) = (1 / (1 − 𝐴))) |
47 | 46 | eqcomd 2616 | . 2 ⊢ (𝜑 → (1 / (1 − 𝐴)) = Σ𝑘 ∈ ℕ0 (𝐴↑𝑘)) |
48 | 41, 43, 47 | 3brtr4d 4615 | 1 ⊢ (𝜑 → (exp‘𝐴) ≤ (1 / (1 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 ℕ0cn0 11169 seqcseq 12663 ↑cexp 12722 !cfa 12922 abscabs 13822 ⇝ cli 14063 Σcsu 14264 expce 14631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 |
This theorem is referenced by: birthdaylem3 24480 logdiflbnd 24521 emcllem2 24523 |
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