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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 12268 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 1nn0 11901 | . . . 4 ⊢ 1 ∈ ℕ0 | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 1 ∈ ℕ0) |
4 | df-2 11688 | . . 3 ⊢ 2 = (1 + 1) | |
5 | rpcn 12387 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
6 | 0nn0 11900 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℕ0) |
8 | 1e0p1 12128 | . . . . 5 ⊢ 1 = (0 + 1) | |
9 | 0z 11980 | . . . . . 6 ⊢ 0 ∈ ℤ | |
10 | eqid 2798 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
11 | 10 | eftval 15422 | . . . . . . . 8 ⊢ (0 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = ((𝐴↑0) / (!‘0))) |
12 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = ((𝐴↑0) / (!‘0)) |
13 | eft0val 15457 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
14 | 12, 13 | syl5eq 2845 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = 1) |
15 | 9, 14 | seq1i 13378 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘0) = 1) |
16 | 10 | eftval 15422 | . . . . . . 7 ⊢ (1 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = ((𝐴↑1) / (!‘1))) |
17 | 2, 16 | ax-mp 5 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = ((𝐴↑1) / (!‘1)) |
18 | fac1 13633 | . . . . . . . 8 ⊢ (!‘1) = 1 | |
19 | 18 | oveq2i 7146 | . . . . . . 7 ⊢ ((𝐴↑1) / (!‘1)) = ((𝐴↑1) / 1) |
20 | exp1 13431 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
21 | 20 | oveq1d 7150 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = (𝐴 / 1)) |
22 | div1 11318 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
23 | 21, 22 | eqtrd 2833 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = 𝐴) |
24 | 19, 23 | syl5eq 2845 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) |
25 | 17, 24 | syl5eq 2845 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = 𝐴) |
26 | 1, 7, 8, 15, 25 | seqp1d 13381 | . . . 4 ⊢ (𝐴 ∈ ℂ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) = (1 + 𝐴)) |
27 | 5, 26 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) = (1 + 𝐴)) |
28 | 2nn0 11902 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
29 | 10 | eftval 15422 | . . . . . 6 ⊢ (2 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘2) = ((𝐴↑2) / (!‘2))) |
30 | 28, 29 | ax-mp 5 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘2) = ((𝐴↑2) / (!‘2)) |
31 | fac2 13635 | . . . . . 6 ⊢ (!‘2) = 2 | |
32 | 31 | oveq2i 7146 | . . . . 5 ⊢ ((𝐴↑2) / (!‘2)) = ((𝐴↑2) / 2) |
33 | 30, 32 | eqtri 2821 | . . . 4 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘2) = ((𝐴↑2) / 2) |
34 | 33 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘2) = ((𝐴↑2) / 2)) |
35 | 1, 3, 4, 27, 34 | seqp1d 13381 | . 2 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘2) = ((1 + 𝐴) + ((𝐴↑2) / 2))) |
36 | id 22 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ+) | |
37 | 28 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 2 ∈ ℕ0) |
38 | 10, 36, 37 | effsumlt 15456 | . 2 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘2) < (exp‘𝐴)) |
39 | 35, 38 | eqbrtrrd 5054 | 1 ⊢ (𝐴 ∈ ℝ+ → ((1 + 𝐴) + ((𝐴↑2) / 2)) < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 < clt 10664 / cdiv 11286 2c2 11680 ℕ0cn0 11885 ℝ+crp 12377 seqcseq 13364 ↑cexp 13425 !cfa 13629 expce 15407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-fac 13630 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 |
This theorem is referenced by: cxp2limlem 25561 pntpbnd1a 26169 |
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