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Mirrors > Home > MPE Home > Th. List > efgt1p | Structured version Visualization version GIF version |
Description: The exponential of a positive real number is greater than 1 plus that number. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
efgt1p | ⊢ (𝐴 ∈ ℝ+ → (1 + 𝐴) < (exp‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpcn 12398 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
2 | nn0uz 12279 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 0nn0 11911 | . . . 4 ⊢ 0 ∈ ℕ0 | |
4 | 1e0p1 12139 | . . . 4 ⊢ 1 = (0 + 1) | |
5 | 0z 11991 | . . . . 5 ⊢ 0 ∈ ℤ | |
6 | eqid 2821 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
7 | 6 | eftval 15429 | . . . . . . 7 ⊢ (0 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = ((𝐴↑0) / (!‘0))) |
8 | 3, 7 | ax-mp 5 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = ((𝐴↑0) / (!‘0)) |
9 | eft0val 15464 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
10 | 8, 9 | syl5eq 2868 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = 1) |
11 | 5, 10 | seq1i 13382 | . . . 4 ⊢ (𝐴 ∈ ℂ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘0) = 1) |
12 | 1nn0 11912 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
13 | 6 | eftval 15429 | . . . . . 6 ⊢ (1 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = ((𝐴↑1) / (!‘1))) |
14 | 12, 13 | ax-mp 5 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = ((𝐴↑1) / (!‘1)) |
15 | fac1 13636 | . . . . . . 7 ⊢ (!‘1) = 1 | |
16 | 15 | oveq2i 7166 | . . . . . 6 ⊢ ((𝐴↑1) / (!‘1)) = ((𝐴↑1) / 1) |
17 | exp1 13434 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
18 | 17 | oveq1d 7170 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = (𝐴 / 1)) |
19 | div1 11328 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
20 | 18, 19 | eqtrd 2856 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = 𝐴) |
21 | 16, 20 | syl5eq 2868 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) |
22 | 14, 21 | syl5eq 2868 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = 𝐴) |
23 | 2, 3, 4, 11, 22 | seqp1i 13385 | . . 3 ⊢ (𝐴 ∈ ℂ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) = (1 + 𝐴)) |
24 | 1, 23 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) = (1 + 𝐴)) |
25 | id 22 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ+) | |
26 | 12 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 1 ∈ ℕ0) |
27 | 6, 25, 26 | effsumlt 15463 | . 2 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) < (exp‘𝐴)) |
28 | 24, 27 | eqbrtrrd 5089 | 1 ⊢ (𝐴 ∈ ℝ+ → (1 + 𝐴) < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 0cc0 10536 1c1 10537 + caddc 10539 < clt 10674 / cdiv 11296 ℕ0cn0 11896 ℝ+crp 12388 seqcseq 13368 ↑cexp 13428 !cfa 13632 expce 15414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-ico 12743 df-fz 12892 df-fzo 13033 df-fl 13161 df-seq 13369 df-exp 13429 df-fac 13633 df-hash 13690 df-shft 14425 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-limsup 14827 df-clim 14844 df-rlim 14845 df-sum 15042 df-ef 15420 |
This theorem is referenced by: efgt1 15468 reeff1olem 25033 logdivlti 25202 logdifbnd 25570 emcllem4 25575 harmonicbnd4 25587 |
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