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Mirrors > Home > ILE Home > Th. List > egt2lt3 | GIF version |
Description: Euler's constant e = 2.71828... is bounded by 2 and 3. (Contributed by NM, 28-Nov-2008.) (Revised by Jim Kingdon, 7-Jan-2023.) |
Ref | Expression |
---|---|
egt2lt3 | ⊢ (2 < e ∧ e < 3) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2189 | . . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛))) = (𝑛 ∈ ℕ ↦ (2 · ((1 / 2)↑𝑛))) | |
2 | eqid 2189 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛))) | |
3 | 1, 2 | ege2le3 11711 | . . . 4 ⊢ (2 ≤ e ∧ e ≤ 3) |
4 | 3 | simpli 111 | . . 3 ⊢ 2 ≤ e |
5 | 2z 9311 | . . . . 5 ⊢ 2 ∈ ℤ | |
6 | zq 9656 | . . . . 5 ⊢ (2 ∈ ℤ → 2 ∈ ℚ) | |
7 | eirrap 11817 | . . . . 5 ⊢ (2 ∈ ℚ → e # 2) | |
8 | 5, 6, 7 | mp2b 8 | . . . 4 ⊢ e # 2 |
9 | ere 11710 | . . . . . 6 ⊢ e ∈ ℝ | |
10 | 9 | recni 7999 | . . . . 5 ⊢ e ∈ ℂ |
11 | 2cn 9020 | . . . . 5 ⊢ 2 ∈ ℂ | |
12 | apsym 8593 | . . . . 5 ⊢ ((e ∈ ℂ ∧ 2 ∈ ℂ) → (e # 2 ↔ 2 # e)) | |
13 | 10, 11, 12 | mp2an 426 | . . . 4 ⊢ (e # 2 ↔ 2 # e) |
14 | 8, 13 | mpbi 145 | . . 3 ⊢ 2 # e |
15 | 2re 9019 | . . . 4 ⊢ 2 ∈ ℝ | |
16 | ltleap 8619 | . . . 4 ⊢ ((2 ∈ ℝ ∧ e ∈ ℝ) → (2 < e ↔ (2 ≤ e ∧ 2 # e))) | |
17 | 15, 9, 16 | mp2an 426 | . . 3 ⊢ (2 < e ↔ (2 ≤ e ∧ 2 # e)) |
18 | 4, 14, 17 | mpbir2an 944 | . 2 ⊢ 2 < e |
19 | 3 | simpri 113 | . . 3 ⊢ e ≤ 3 |
20 | 3z 9312 | . . . 4 ⊢ 3 ∈ ℤ | |
21 | zq 9656 | . . . 4 ⊢ (3 ∈ ℤ → 3 ∈ ℚ) | |
22 | eirrap 11817 | . . . 4 ⊢ (3 ∈ ℚ → e # 3) | |
23 | 20, 21, 22 | mp2b 8 | . . 3 ⊢ e # 3 |
24 | 3re 9023 | . . . 4 ⊢ 3 ∈ ℝ | |
25 | ltleap 8619 | . . . 4 ⊢ ((e ∈ ℝ ∧ 3 ∈ ℝ) → (e < 3 ↔ (e ≤ 3 ∧ e # 3))) | |
26 | 9, 24, 25 | mp2an 426 | . . 3 ⊢ (e < 3 ↔ (e ≤ 3 ∧ e # 3)) |
27 | 19, 23, 26 | mpbir2an 944 | . 2 ⊢ e < 3 |
28 | 18, 27 | pm3.2i 272 | 1 ⊢ (2 < e ∧ e < 3) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∈ wcel 2160 class class class wbr 4018 ↦ cmpt 4079 ‘cfv 5235 (class class class)co 5896 ℂcc 7839 ℝcr 7840 1c1 7842 · cmul 7846 < clt 8022 ≤ cle 8023 # cap 8568 / cdiv 8659 ℕcn 8949 2c2 9000 3c3 9001 ℕ0cn0 9206 ℤcz 9283 ℚcq 9649 ↑cexp 10550 !cfa 10737 eceu 11683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 ax-pre-mulext 7959 ax-arch 7960 ax-caucvg 7961 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-isom 5244 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-irdg 6395 df-frec 6416 df-1o 6441 df-oadd 6445 df-er 6559 df-en 6767 df-dom 6768 df-fin 6769 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 df-div 8660 df-inn 8950 df-2 9008 df-3 9009 df-4 9010 df-n0 9207 df-z 9284 df-uz 9559 df-q 9650 df-rp 9684 df-ico 9924 df-fz 10039 df-fzo 10173 df-seqfrec 10477 df-exp 10551 df-fac 10738 df-bc 10760 df-ihash 10788 df-shft 10856 df-cj 10883 df-re 10884 df-im 10885 df-rsqrt 11039 df-abs 11040 df-clim 11319 df-sumdc 11394 df-ef 11688 df-e 11689 |
This theorem is referenced by: epos 11820 ene1 11824 eap1 11825 reeff1o 14651 |
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