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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 strictly bounded below by 2 and above by 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 2735 | . . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛))) = (𝑛 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛))) | |
2 | eqid 2735 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛))) | |
3 | 1, 2 | ege2le3 16123 | . . . 4 ⊢ (2 ≤ e ∧ e ≤ 3) |
4 | 3 | simpli 483 | . . 3 ⊢ 2 ≤ e |
5 | eirr 16238 | . . . . . 6 ⊢ e ∉ ℚ | |
6 | 5 | neli 3046 | . . . . 5 ⊢ ¬ e ∈ ℚ |
7 | nnq 13002 | . . . . 5 ⊢ (e ∈ ℕ → e ∈ ℚ) | |
8 | 6, 7 | mto 197 | . . . 4 ⊢ ¬ e ∈ ℕ |
9 | 2nn 12337 | . . . . . 6 ⊢ 2 ∈ ℕ | |
10 | eleq1 2827 | . . . . . 6 ⊢ (e = 2 → (e ∈ ℕ ↔ 2 ∈ ℕ)) | |
11 | 9, 10 | mpbiri 258 | . . . . 5 ⊢ (e = 2 → e ∈ ℕ) |
12 | 11 | necon3bi 2965 | . . . 4 ⊢ (¬ e ∈ ℕ → e ≠ 2) |
13 | 8, 12 | ax-mp 5 | . . 3 ⊢ e ≠ 2 |
14 | 2re 12338 | . . . 4 ⊢ 2 ∈ ℝ | |
15 | ere 16122 | . . . 4 ⊢ e ∈ ℝ | |
16 | 14, 15 | ltleni 11377 | . . 3 ⊢ (2 < e ↔ (2 ≤ e ∧ e ≠ 2)) |
17 | 4, 13, 16 | mpbir2an 711 | . 2 ⊢ 2 < e |
18 | 3 | simpri 485 | . . 3 ⊢ e ≤ 3 |
19 | 3nn 12343 | . . . . . 6 ⊢ 3 ∈ ℕ | |
20 | eleq1 2827 | . . . . . 6 ⊢ (3 = e → (3 ∈ ℕ ↔ e ∈ ℕ)) | |
21 | 19, 20 | mpbii 233 | . . . . 5 ⊢ (3 = e → e ∈ ℕ) |
22 | 21 | necon3bi 2965 | . . . 4 ⊢ (¬ e ∈ ℕ → 3 ≠ e) |
23 | 8, 22 | ax-mp 5 | . . 3 ⊢ 3 ≠ e |
24 | 3re 12344 | . . . 4 ⊢ 3 ∈ ℝ | |
25 | 15, 24 | ltleni 11377 | . . 3 ⊢ (e < 3 ↔ (e ≤ 3 ∧ 3 ≠ e)) |
26 | 18, 23, 25 | mpbir2an 711 | . 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 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 1c1 11154 · cmul 11158 < clt 11293 ≤ cle 11294 / cdiv 11918 ℕcn 12264 2c2 12319 3c3 12320 ℕ0cn0 12524 ℚcq 12988 ↑cexp 14099 !cfa 14309 eceu 16095 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-ico 13390 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-e 16101 |
This theorem is referenced by: epos 16240 ene1 16243 cxploglim2 27037 harmonicbnd3 27066 bposlem7 27349 bposlem9 27351 chebbnd1lem2 27529 chebbnd1lem3 27530 chebbnd1 27531 dchrvmasumlema 27559 mulog2sumlem2 27594 pntpbnd1a 27644 pntpbnd2 27646 pntlemb 27656 pntlemk 27665 hgt750lem 34645 subfacval3 35174 aks4d1p1p7 42056 etransclem23 46213 |
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