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Mirrors > Home > MPE Home > Th. List > eflt | Structured version Visualization version GIF version |
Description: The exponential function on the reals is strictly increasing. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 17-Jul-2014.) |
Ref | Expression |
---|---|
eflt | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (exp‘𝐴) < (exp‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1532 | . 2 ⊢ ⊤ | |
2 | fveq2 6664 | . . 3 ⊢ (𝑥 = 𝑦 → (exp‘𝑥) = (exp‘𝑦)) | |
3 | fveq2 6664 | . . 3 ⊢ (𝑥 = 𝐴 → (exp‘𝑥) = (exp‘𝐴)) | |
4 | fveq2 6664 | . . 3 ⊢ (𝑥 = 𝐵 → (exp‘𝑥) = (exp‘𝐵)) | |
5 | ssid 3988 | . . 3 ⊢ ℝ ⊆ ℝ | |
6 | reefcl 15430 | . . . 4 ⊢ (𝑥 ∈ ℝ → (exp‘𝑥) ∈ ℝ) | |
7 | 6 | adantl 482 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (exp‘𝑥) ∈ ℝ) |
8 | simp2 1129 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℝ) | |
9 | simp1 1128 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℝ) | |
10 | 8, 9 | resubcld 11057 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℝ) |
11 | posdif 11122 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ 0 < (𝑦 − 𝑥))) | |
12 | 11 | biimp3a 1460 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 < (𝑦 − 𝑥)) |
13 | 10, 12 | elrpd 12418 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℝ+) |
14 | efgt1 15459 | . . . . . . . 8 ⊢ ((𝑦 − 𝑥) ∈ ℝ+ → 1 < (exp‘(𝑦 − 𝑥))) | |
15 | 13, 14 | syl 17 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 1 < (exp‘(𝑦 − 𝑥))) |
16 | 9 | reefcld 15431 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (exp‘𝑥) ∈ ℝ) |
17 | 10 | reefcld 15431 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (exp‘(𝑦 − 𝑥)) ∈ ℝ) |
18 | efgt0 15446 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 0 < (exp‘𝑥)) | |
19 | 9, 18 | syl 17 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 0 < (exp‘𝑥)) |
20 | ltmulgt11 11489 | . . . . . . . 8 ⊢ (((exp‘𝑥) ∈ ℝ ∧ (exp‘(𝑦 − 𝑥)) ∈ ℝ ∧ 0 < (exp‘𝑥)) → (1 < (exp‘(𝑦 − 𝑥)) ↔ (exp‘𝑥) < ((exp‘𝑥) · (exp‘(𝑦 − 𝑥))))) | |
21 | 16, 17, 19, 20 | syl3anc 1363 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (1 < (exp‘(𝑦 − 𝑥)) ↔ (exp‘𝑥) < ((exp‘𝑥) · (exp‘(𝑦 − 𝑥))))) |
22 | 15, 21 | mpbid 233 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (exp‘𝑥) < ((exp‘𝑥) · (exp‘(𝑦 − 𝑥)))) |
23 | 9 | recnd 10658 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑥 ∈ ℂ) |
24 | 10 | recnd 10658 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑦 − 𝑥) ∈ ℂ) |
25 | efadd 15437 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℂ ∧ (𝑦 − 𝑥) ∈ ℂ) → (exp‘(𝑥 + (𝑦 − 𝑥))) = ((exp‘𝑥) · (exp‘(𝑦 − 𝑥)))) | |
26 | 23, 24, 25 | syl2anc 584 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (exp‘(𝑥 + (𝑦 − 𝑥))) = ((exp‘𝑥) · (exp‘(𝑦 − 𝑥)))) |
27 | 8 | recnd 10658 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → 𝑦 ∈ ℂ) |
28 | 23, 27 | pncan3d 10989 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (𝑥 + (𝑦 − 𝑥)) = 𝑦) |
29 | 28 | fveq2d 6668 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (exp‘(𝑥 + (𝑦 − 𝑥))) = (exp‘𝑦)) |
30 | 26, 29 | eqtr3d 2858 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → ((exp‘𝑥) · (exp‘(𝑦 − 𝑥))) = (exp‘𝑦)) |
31 | 22, 30 | breqtrd 5084 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 < 𝑦) → (exp‘𝑥) < (exp‘𝑦)) |
32 | 31 | 3expia 1113 | . . . 4 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 → (exp‘𝑥) < (exp‘𝑦))) |
33 | 32 | adantl 482 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 < 𝑦 → (exp‘𝑥) < (exp‘𝑦))) |
34 | 2, 3, 4, 5, 7, 33 | ltord1 11155 | . 2 ⊢ ((⊤ ∧ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) → (𝐴 < 𝐵 ↔ (exp‘𝐴) < (exp‘𝐵))) |
35 | 1, 34 | mpan 686 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ (exp‘𝐴) < (exp‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ⊤wtru 1529 ∈ wcel 2105 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 − cmin 10859 ℝ+crp 12379 expce 15405 |
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 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 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 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 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 3497 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 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-pm 8399 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-inf 8896 df-oi 8963 df-card 9357 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 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-rp 12380 df-ico 12734 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-fac 13624 df-bc 13653 df-hash 13681 df-shft 14416 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-limsup 14818 df-clim 14835 df-rlim 14836 df-sum 15033 df-ef 15411 |
This theorem is referenced by: efle 15461 reefiso 24965 logdivlti 25130 divlogrlim 25145 cxplt 25204 birthday 25460 cxploglim 25483 bposlem6 25793 bposlem9 25796 pntpbnd1a 26089 pntibndlem2 26095 pntlemb 26101 ostth2lem3 26139 ostth2 26141 |
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