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Mirrors > Home > MPE Home > Th. List > efgt1p2 | Structured version Visualization version GIF version |
Description: The exponential function of a positive real number is greater than 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 11598 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 1nn0 11185 | . . 3 ⊢ 1 ∈ ℕ0 | |
3 | df-2 10956 | . . 3 ⊢ 2 = (1 + 1) | |
4 | rpcn 11717 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
5 | 0nn0 11184 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
6 | 1e0p1 11428 | . . . . 5 ⊢ 1 = (0 + 1) | |
7 | 0z 11265 | . . . . . 6 ⊢ 0 ∈ ℤ | |
8 | eqid 2610 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
9 | 8 | eftval 14646 | . . . . . . . 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 14681 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
12 | 10, 11 | syl5eq 2656 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = 1) |
13 | 7, 12 | seq1i 12677 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘0) = 1) |
14 | 8 | eftval 14646 | . . . . . . 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 12926 | . . . . . . . 8 ⊢ (!‘1) = 1 | |
17 | 16 | oveq2i 6560 | . . . . . . 7 ⊢ ((𝐴↑1) / (!‘1)) = ((𝐴↑1) / 1) |
18 | exp1 12728 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
19 | 18 | oveq1d 6564 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = (𝐴 / 1)) |
20 | div1 10595 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
21 | 19, 20 | eqtrd 2644 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = 𝐴) |
22 | 17, 21 | syl5eq 2656 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) |
23 | 15, 22 | syl5eq 2656 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = 𝐴) |
24 | 1, 5, 6, 13, 23 | seqp1i 12679 | . . . 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 11186 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
27 | 8 | eftval 14646 | . . . . . 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 12928 | . . . . . 6 ⊢ (!‘2) = 2 | |
30 | 29 | oveq2i 6560 | . . . . 5 ⊢ ((𝐴↑2) / (!‘2)) = ((𝐴↑2) / 2) |
31 | 28, 30 | eqtri 2632 | . . . 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 12679 | . 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 14680 | . 2 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘2) < (exp‘𝐴)) |
37 | 33, 36 | eqbrtrrd 4607 | 1 ⊢ (𝐴 ∈ ℝ+ → ((1 + 𝐴) + ((𝐴↑2) / 2)) < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 / cdiv 10563 2c2 10947 ℕ0cn0 11169 ℝ+crp 11708 seqcseq 12663 ↑cexp 12722 !cfa 12922 expce 14631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 |
This theorem is referenced by: cxp2limlem 24502 pntpbnd1a 25074 |
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