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Mirrors > Home > MPE Home > Th. List > efgt1p2 | Structured version Visualization version GIF version |
Description: The exponential of a positive real number is greater than the sum of the first three terms of the series expansion. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
efgt1p2 | ⊢ (𝐴 ∈ ℝ+ → ((1 + 𝐴) + ((𝐴↑2) / 2)) < (exp‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12005 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 1nn0 11637 | . . 3 ⊢ 1 ∈ ℕ0 | |
3 | df-2 11415 | . . 3 ⊢ 2 = (1 + 1) | |
4 | rpcn 12125 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
5 | 0nn0 11636 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
6 | 1e0p1 11865 | . . . . 5 ⊢ 1 = (0 + 1) | |
7 | 0z 11716 | . . . . . 6 ⊢ 0 ∈ ℤ | |
8 | eqid 2826 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
9 | 8 | eftval 15180 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = ((𝐴↑0) / (!‘0))) |
10 | 5, 9 | ax-mp 5 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = ((𝐴↑0) / (!‘0)) |
11 | eft0val 15215 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
12 | 10, 11 | syl5eq 2874 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = 1) |
13 | 7, 12 | seq1i 13110 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘0) = 1) |
14 | 8 | eftval 15180 | . . . . . . 7 ⊢ (1 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = ((𝐴↑1) / (!‘1))) |
15 | 2, 14 | ax-mp 5 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = ((𝐴↑1) / (!‘1)) |
16 | fac1 13358 | . . . . . . . 8 ⊢ (!‘1) = 1 | |
17 | 16 | oveq2i 6917 | . . . . . . 7 ⊢ ((𝐴↑1) / (!‘1)) = ((𝐴↑1) / 1) |
18 | exp1 13161 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
19 | 18 | oveq1d 6921 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = (𝐴 / 1)) |
20 | div1 11042 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
21 | 19, 20 | eqtrd 2862 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = 𝐴) |
22 | 17, 21 | syl5eq 2874 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) |
23 | 15, 22 | syl5eq 2874 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = 𝐴) |
24 | 1, 5, 6, 13, 23 | seqp1i 13112 | . . . 4 ⊢ (𝐴 ∈ ℂ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) = (1 + 𝐴)) |
25 | 4, 24 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) = (1 + 𝐴)) |
26 | 2nn0 11638 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
27 | 8 | eftval 15180 | . . . . . 6 ⊢ (2 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘2) = ((𝐴↑2) / (!‘2))) |
28 | 26, 27 | ax-mp 5 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘2) = ((𝐴↑2) / (!‘2)) |
29 | fac2 13360 | . . . . . 6 ⊢ (!‘2) = 2 | |
30 | 29 | oveq2i 6917 | . . . . 5 ⊢ ((𝐴↑2) / (!‘2)) = ((𝐴↑2) / 2) |
31 | 28, 30 | eqtri 2850 | . . . 4 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘2) = ((𝐴↑2) / 2) |
32 | 31 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘2) = ((𝐴↑2) / 2)) |
33 | 1, 2, 3, 25, 32 | seqp1i 13112 | . 2 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘2) = ((1 + 𝐴) + ((𝐴↑2) / 2))) |
34 | id 22 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ+) | |
35 | 26 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 2 ∈ ℕ0) |
36 | 8, 34, 35 | effsumlt 15214 | . 2 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘2) < (exp‘𝐴)) |
37 | 33, 36 | eqbrtrrd 4898 | 1 ⊢ (𝐴 ∈ ℝ+ → ((1 + 𝐴) + ((𝐴↑2) / 2)) < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 class class class wbr 4874 ↦ cmpt 4953 ‘cfv 6124 (class class class)co 6906 ℂcc 10251 0cc0 10253 1c1 10254 + caddc 10256 < clt 10392 / cdiv 11010 2c2 11407 ℕ0cn0 11619 ℝ+crp 12113 seqcseq 13096 ↑cexp 13155 !cfa 13354 expce 15165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 ax-addf 10332 ax-mulf 10333 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-sup 8618 df-inf 8619 df-oi 8685 df-card 9079 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-z 11706 df-uz 11970 df-rp 12114 df-ico 12470 df-fz 12621 df-fzo 12762 df-fl 12889 df-seq 13097 df-exp 13156 df-fac 13355 df-hash 13412 df-shft 14185 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-limsup 14580 df-clim 14597 df-rlim 14598 df-sum 14795 df-ef 15171 |
This theorem is referenced by: cxp2limlem 25116 pntpbnd1a 25688 |
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