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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 12387 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
2 | nn0uz 12268 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 0nn0 11900 | . . . 4 ⊢ 0 ∈ ℕ0 | |
4 | 1e0p1 12128 | . . . 4 ⊢ 1 = (0 + 1) | |
5 | 0z 11980 | . . . . 5 ⊢ 0 ∈ ℤ | |
6 | eqid 2818 | . . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
7 | 6 | eftval 15418 | . . . . . . 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 15453 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
10 | 8, 9 | syl5eq 2865 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘0) = 1) |
11 | 5, 10 | seq1i 13371 | . . . 4 ⊢ (𝐴 ∈ ℂ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘0) = 1) |
12 | 1nn0 11901 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
13 | 6 | eftval 15418 | . . . . . 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 13625 | . . . . . . 7 ⊢ (!‘1) = 1 | |
16 | 15 | oveq2i 7156 | . . . . . 6 ⊢ ((𝐴↑1) / (!‘1)) = ((𝐴↑1) / 1) |
17 | exp1 13423 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
18 | 17 | oveq1d 7160 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = (𝐴 / 1)) |
19 | div1 11317 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
20 | 18, 19 | eqtrd 2853 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / 1) = 𝐴) |
21 | 16, 20 | syl5eq 2865 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) |
22 | 14, 21 | syl5eq 2865 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘1) = 𝐴) |
23 | 2, 3, 4, 11, 22 | seqp1i 13374 | . . 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 15452 | . 2 ⊢ (𝐴 ∈ ℝ+ → (seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))))‘1) < (exp‘𝐴)) |
28 | 24, 27 | eqbrtrrd 5081 | 1 ⊢ (𝐴 ∈ ℝ+ → (1 + 𝐴) < (exp‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 < clt 10663 / cdiv 11285 ℕ0cn0 11885 ℝ+crp 12377 seqcseq 13357 ↑cexp 13417 !cfa 13621 expce 15403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-fac 13622 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 |
This theorem is referenced by: efgt1 15457 reeff1olem 24961 logdivlti 25130 logdifbnd 25498 emcllem4 25503 harmonicbnd4 25515 |
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