egt2lt3 - Metamath Proof Explorer

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Theorem egt2lt3 16176

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.)

**Assertion**
Ref	Expression
egt2lt3	⊢ (2 < e ∧ e < 3)

**Proof of Theorem egt2lt3**
Step	Hyp	Ref	Expression
1		eqid 2728	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛))) = (𝑛 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛)))
2		eqid 2728	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ ↦ (1 / (!‘𝑛))) = (𝑛 ∈ ℕ₀ ↦ (1 / (!‘𝑛)))
3	1, 2	ege2le3 16060	. . . 4 ⊢ (2 ≤ e ∧ e ≤ 3)
4	3	simpli 483	. . 3 ⊢ 2 ≤ e
5		eirr 16175	. . . . . 6 ⊢ e ∉ ℚ
6	5	neli 3044	. . . . 5 ⊢ ¬ e ∈ ℚ
7		nnq 12970	. . . . 5 ⊢ (e ∈ ℕ → e ∈ ℚ)
8	6, 7	mto 196	. . . 4 ⊢ ¬ e ∈ ℕ
9		2nn 12309	. . . . . 6 ⊢ 2 ∈ ℕ
10		eleq1 2817	. . . . . 6 ⊢ (e = 2 → (e ∈ ℕ ↔ 2 ∈ ℕ))
11	9, 10	mpbiri 258	. . . . 5 ⊢ (e = 2 → e ∈ ℕ)
12	11	necon3bi 2963	. . . 4 ⊢ (¬ e ∈ ℕ → e ≠ 2)
13	8, 12	ax-mp 5	. . 3 ⊢ e ≠ 2
14		2re 12310	. . . 4 ⊢ 2 ∈ ℝ
15		ere 16059	. . . 4 ⊢ e ∈ ℝ
16	14, 15	ltleni 11356	. . 3 ⊢ (2 < e ↔ (2 ≤ e ∧ e ≠ 2))
17	4, 13, 16	mpbir2an 710	. 2 ⊢ 2 < e
18	3	simpri 485	. . 3 ⊢ e ≤ 3
19		3nn 12315	. . . . . 6 ⊢ 3 ∈ ℕ
20		eleq1 2817	. . . . . 6 ⊢ (3 = e → (3 ∈ ℕ ↔ e ∈ ℕ))
21	19, 20	mpbii 232	. . . . 5 ⊢ (3 = e → e ∈ ℕ)
22	21	necon3bi 2963	. . . 4 ⊢ (¬ e ∈ ℕ → 3 ≠ e)
23	8, 22	ax-mp 5	. . 3 ⊢ 3 ≠ e
24		3re 12316	. . . 4 ⊢ 3 ∈ ℝ
25	15, 24	ltleni 11356	. . 3 ⊢ (e < 3 ↔ (e ≤ 3 ∧ 3 ≠ e))
26	18, 23, 25	mpbir2an 710	. 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 1534 ∈ wcel 2099 ≠ wne 2936 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 1c1 11133 · cmul 11137 < clt 11272 ≤ cle 11273 / cdiv 11895 ℕcn 12236 2c2 12291 3c3 12292 ℕ₀cn0 12496 ℚcq 12956 ↑cexp 14052 !cfa 14258 eceu 16032

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-n0 12497 df-z 12583 df-uz 12847 df-q 12957 df-rp 13001 df-ico 13356 df-fz 13511 df-fzo 13654 df-fl 13783 df-seq 13993 df-exp 14053 df-fac 14259 df-bc 14288 df-hash 14316 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15441 df-clim 15458 df-rlim 15459 df-sum 15659 df-ef 16037 df-e 16038

This theorem is referenced by: epos 16177 ene1 16180 cxploglim2 26904 harmonicbnd3 26933 bposlem7 27216 bposlem9 27218 chebbnd1lem2 27396 chebbnd1lem3 27397 chebbnd1 27398 dchrvmasumlema 27426 mulog2sumlem2 27461 pntpbnd1a 27511 pntpbnd2 27513 pntlemb 27523 pntlemk 27532 hgt750lem 34277 subfacval3 34793 aks4d1p1p7 41539 etransclem23 45639

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