logdivlt - Metamath Proof Explorer

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Theorem logdivlt 26664

Description: The log𝑥 / 𝑥 function is strictly decreasing on the reals greater than e. (Contributed by Mario Carneiro, 14-Mar-2014.)

**Assertion**
Ref	Expression
logdivlt	⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 ↔ ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))

**Proof of Theorem logdivlt**
Step	Hyp	Ref	Expression
1		logdivlti 26663	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐴 < 𝐵) → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))
2	1	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
3	2	3expa 1118	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ e ≤ 𝐴) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
4	3	an32s 652	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
5	4	adantrr 717	. 2 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
6		fveq2 6905	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (log‘𝐴) = (log‘𝐵))
7		id 22	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
8	6, 7	oveq12d 7450	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((log‘𝐴) / 𝐴) = ((log‘𝐵) / 𝐵))
9	8	eqcomd 2742	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴))
10	9	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 = 𝐵 → ((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴)))
11		logdivlti 26663	. . . . . . . . . 10 ⊢ (((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ e ≤ 𝐵) ∧ 𝐵 < 𝐴) → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))
12	11	ex 412	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ e ≤ 𝐵) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
13	12	3expa 1118	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ e ≤ 𝐵) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
14	13	an32s 652	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ ∧ e ≤ 𝐵) ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
15	14	adantrr 717	. . . . . 6 ⊢ (((𝐵 ∈ ℝ ∧ e ≤ 𝐵) ∧ (𝐴 ∈ ℝ ∧ e ≤ 𝐴)) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
16	15	ancoms 458	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
17	10, 16	orim12d 966	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → (((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴) ∨ ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))))
18	17	con3d 152	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (¬ (((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴) ∨ ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)) → ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
19		simpl 482	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 𝐵 ∈ ℝ)
20		epos 16244	. . . . . . . 8 ⊢ 0 < e
21		0re 11264	. . . . . . . . 9 ⊢ 0 ∈ ℝ
22		ere 16126	. . . . . . . . 9 ⊢ e ∈ ℝ
23		ltletr 11354	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ e ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < e ∧ e ≤ 𝐵) → 0 < 𝐵))
24	21, 22, 23	mp3an12 1452	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → ((0 < e ∧ e ≤ 𝐵) → 0 < 𝐵))
25	20, 24	mpani 696	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (e ≤ 𝐵 → 0 < 𝐵))
26	25	imp 406	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 0 < 𝐵)
27	19, 26	elrpd 13075	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 𝐵 ∈ ℝ⁺)
28		relogcl 26618	. . . . . 6 ⊢ (𝐵 ∈ ℝ⁺ → (log‘𝐵) ∈ ℝ)
29		rerpdivcl 13066	. . . . . 6 ⊢ (((log‘𝐵) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((log‘𝐵) / 𝐵) ∈ ℝ)
30	28, 29	mpancom 688	. . . . 5 ⊢ (𝐵 ∈ ℝ⁺ → ((log‘𝐵) / 𝐵) ∈ ℝ)
31	27, 30	syl 17	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → ((log‘𝐵) / 𝐵) ∈ ℝ)
32		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 𝐴 ∈ ℝ)
33		ltletr 11354	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ e ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < e ∧ e ≤ 𝐴) → 0 < 𝐴))
34	21, 22, 33	mp3an12 1452	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((0 < e ∧ e ≤ 𝐴) → 0 < 𝐴))
35	20, 34	mpani 696	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (e ≤ 𝐴 → 0 < 𝐴))
36	35	imp 406	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 0 < 𝐴)
37	32, 36	elrpd 13075	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
38		relogcl 26618	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
39		rerpdivcl 13066	. . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → ((log‘𝐴) / 𝐴) ∈ ℝ)
40	38, 39	mpancom 688	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) / 𝐴) ∈ ℝ)
41	37, 40	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → ((log‘𝐴) / 𝐴) ∈ ℝ)
42		axlttri 11333	. . . 4 ⊢ ((((log‘𝐵) / 𝐵) ∈ ℝ ∧ ((log‘𝐴) / 𝐴) ∈ ℝ) → (((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴) ↔ ¬ (((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴) ∨ ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))))
43	31, 41, 42	syl2anr 597	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴) ↔ ¬ (((log‘𝐵) / 𝐵) = ((log‘𝐴) / 𝐴) ∨ ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))))
44		axlttri 11333	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
45	44	ad2ant2r 747	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
46	18, 43, 45	3imtr4d 294	. 2 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴) → 𝐴 < 𝐵))
47	5, 46	impbid 212	1 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 ↔ ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 0cc0 11156 < clt 11296 ≤ cle 11297 / cdiv 11921 ℝ⁺crp 13035 eceu 16099 logclog 26597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-e 16105 df-sin 16106 df-cos 16107 df-pi 16109 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-log 26599

This theorem is referenced by: logdivle 26665 bposlem7 27335 chebbnd1lem2 27515 chebbnd1lem3 27516 pntpbnd1a 27630

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