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Mirrors > Home > MPE Home > Th. List > egt2lt3 | Structured version Visualization version GIF version |
Description: Euler's constant e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Mario Carneiro, 29-Apr-2014.) |
Ref | Expression |
---|---|
egt2lt3 | ⊢ (2 < e ∧ e < 3) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2651 | . . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛))) = (𝑛 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛))) | |
2 | eqid 2651 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛))) | |
3 | 1, 2 | ege2le3 14864 | . . . 4 ⊢ (2 ≤ e ∧ e ≤ 3) |
4 | 3 | simpli 473 | . . 3 ⊢ 2 ≤ e |
5 | eirr 14977 | . . . . . 6 ⊢ e ∉ ℚ | |
6 | 5 | neli 2928 | . . . . 5 ⊢ ¬ e ∈ ℚ |
7 | nnq 11839 | . . . . 5 ⊢ (e ∈ ℕ → e ∈ ℚ) | |
8 | 6, 7 | mto 188 | . . . 4 ⊢ ¬ e ∈ ℕ |
9 | 2nn 11223 | . . . . . 6 ⊢ 2 ∈ ℕ | |
10 | eleq1 2718 | . . . . . 6 ⊢ (e = 2 → (e ∈ ℕ ↔ 2 ∈ ℕ)) | |
11 | 9, 10 | mpbiri 248 | . . . . 5 ⊢ (e = 2 → e ∈ ℕ) |
12 | 11 | necon3bi 2849 | . . . 4 ⊢ (¬ e ∈ ℕ → e ≠ 2) |
13 | 8, 12 | ax-mp 5 | . . 3 ⊢ e ≠ 2 |
14 | 2re 11128 | . . . 4 ⊢ 2 ∈ ℝ | |
15 | ere 14863 | . . . 4 ⊢ e ∈ ℝ | |
16 | 14, 15 | ltleni 10193 | . . 3 ⊢ (2 < e ↔ (2 ≤ e ∧ e ≠ 2)) |
17 | 4, 13, 16 | mpbir2an 975 | . 2 ⊢ 2 < e |
18 | 3 | simpri 477 | . . 3 ⊢ e ≤ 3 |
19 | 3nn 11224 | . . . . . 6 ⊢ 3 ∈ ℕ | |
20 | eleq1 2718 | . . . . . 6 ⊢ (3 = e → (3 ∈ ℕ ↔ e ∈ ℕ)) | |
21 | 19, 20 | mpbii 223 | . . . . 5 ⊢ (3 = e → e ∈ ℕ) |
22 | 21 | necon3bi 2849 | . . . 4 ⊢ (¬ e ∈ ℕ → 3 ≠ e) |
23 | 8, 22 | ax-mp 5 | . . 3 ⊢ 3 ≠ e |
24 | 3re 11132 | . . . 4 ⊢ 3 ∈ ℝ | |
25 | 15, 24 | ltleni 10193 | . . 3 ⊢ (e < 3 ↔ (e ≤ 3 ∧ 3 ≠ e)) |
26 | 18, 23, 25 | mpbir2an 975 | . 2 ⊢ e < 3 |
27 | 17, 26 | pm3.2i 470 | 1 ⊢ (2 < e ∧ e < 3) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 class class class wbr 4685 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 1c1 9975 · cmul 9979 < clt 10112 ≤ cle 10113 / cdiv 10722 ℕcn 11058 2c2 11108 3c3 11109 ℕ0cn0 11330 ℚcq 11826 ↑cexp 12900 !cfa 13100 eceu 14837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-pm 7902 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-n0 11331 df-z 11416 df-uz 11726 df-q 11827 df-rp 11871 df-ico 12219 df-fz 12365 df-fzo 12505 df-fl 12633 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-e 14843 |
This theorem is referenced by: epos 14979 ene1 14982 logblog 24575 cxploglim2 24750 emgt0 24778 harmonicbnd3 24779 bposlem7 25060 bposlem9 25062 chebbnd1lem2 25204 chebbnd1lem3 25205 chebbnd1 25206 dchrvmasumlema 25234 mulog2sumlem2 25269 pntpbnd1a 25319 pntpbnd2 25321 pntlemb 25331 pntlemk 25340 hgt750lem 30857 subfacval3 31297 etransclem23 40792 |
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