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Theorem logdivlt 26762

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 26761	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐴 < 𝐵) → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))
2	1	ex 417	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
3	2	3expa 1134	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ e ≤ 𝐴) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
4	3	an32s 664	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
5	4	adantrr 729	. 2 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
6		fveq2 6881	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (log‘𝐴) = (log‘𝐵))
7		id 23	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
8	6, 7	oveq12d 7428	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((log‘𝐴) / 𝐴) = ((log‘𝐵) / 𝐵))
9	8	eqcomd 2767	. . . . . 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 26761	. . . . . . . . . 10 ⊢ (((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ e ≤ 𝐵) ∧ 𝐵 < 𝐴) → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))
12	11	ex 417	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ e ≤ 𝐵) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
13	12	3expa 1134	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ e ≤ 𝐵) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
14	13	an32s 664	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ ∧ e ≤ 𝐵) ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
15	14	adantrr 729	. . . . . 6 ⊢ (((𝐵 ∈ ℝ ∧ e ≤ 𝐵) ∧ (𝐴 ∈ ℝ ∧ e ≤ 𝐴)) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
16	15	ancoms 463	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
17	10, 16	orim12d 979	. . . 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 153	. . 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 487	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 𝐵 ∈ ℝ)
20		epos 16262	. . . . . . . 8 ⊢ 0 < e
21		0re 11209	. . . . . . . . 9 ⊢ 0 ∈ ℝ
22		ere 16142	. . . . . . . . 9 ⊢ e ∈ ℝ
23		ltletr 11301	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ e ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < e ∧ e ≤ 𝐵) → 0 < 𝐵))
24	21, 22, 23	mp3an12 1478	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → ((0 < e ∧ e ≤ 𝐵) → 0 < 𝐵))
25	20, 24	mpani 708	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (e ≤ 𝐵 → 0 < 𝐵))
26	25	imp 411	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 0 < 𝐵)
27	19, 26	elrpd 13056	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 𝐵 ∈ ℝ⁺)
28		relogcl 26716	. . . . . 6 ⊢ (𝐵 ∈ ℝ⁺ → (log‘𝐵) ∈ ℝ)
29		rerpdivcl 13047	. . . . . 6 ⊢ (((log‘𝐵) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((log‘𝐵) / 𝐵) ∈ ℝ)
30	28, 29	mpancom 700	. . . . 5 ⊢ (𝐵 ∈ ℝ⁺ → ((log‘𝐵) / 𝐵) ∈ ℝ)
31	27, 30	syl 18	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → ((log‘𝐵) / 𝐵) ∈ ℝ)
32		simpl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 𝐴 ∈ ℝ)
33		ltletr 11301	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ e ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < e ∧ e ≤ 𝐴) → 0 < 𝐴))
34	21, 22, 33	mp3an12 1478	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((0 < e ∧ e ≤ 𝐴) → 0 < 𝐴))
35	20, 34	mpani 708	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (e ≤ 𝐴 → 0 < 𝐴))
36	35	imp 411	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 0 < 𝐴)
37	32, 36	elrpd 13056	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
38		relogcl 26716	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
39		rerpdivcl 13047	. . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → ((log‘𝐴) / 𝐴) ∈ ℝ)
40	38, 39	mpancom 700	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) / 𝐴) ∈ ℝ)
41	37, 40	syl 18	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → ((log‘𝐴) / 𝐴) ∈ ℝ)
42		axlttri 11280	. . . 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 608	. . 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 11280	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
45	44	ad2ant2r 759	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
46	18, 43, 45	3imtr4d 297	. 2 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴) → 𝐴 < 𝐵))
47	5, 46	impbid 215	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 class class class wbr 5108 ‘cfv 6536 (class class class)co 7410 ℝcr 11098 0cc0 11099 < clt 11242 ≤ cle 11243 / cdiv 11870 ℝ⁺crp 13015 eceu 16115 logclog 26695

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13375 df-ioc 13376 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13683 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15104 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-limsup 15522 df-clim 15539 df-rlim 15540 df-sum 15738 df-ef 16120 df-e 16121 df-sin 16122 df-cos 16123 df-pi 16125 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-mulg 19133 df-cntz 19386 df-cmn 19851 df-psmet 21493 df-xmet 21494 df-met 21495 df-bl 21496 df-mopn 21497 df-fbas 21498 df-fg 21499 df-cnfld 21502 df-top 23030 df-topon 23047 df-topsp 23069 df-bases 23082 df-cld 23155 df-ntr 23156 df-cls 23157 df-nei 23234 df-lp 23272 df-perf 23273 df-cn 23363 df-cnp 23364 df-haus 23451 df-tx 23698 df-hmeo 23891 df-fil 23982 df-fm 24074 df-flim 24075 df-flf 24076 df-xms 24456 df-ms 24457 df-tms 24458 df-cncf 25016 df-limc 26004 df-dv 26005 df-log 26697

This theorem is referenced by: logdivle 26763 bposlem7 27430 chebbnd1lem2 27610 chebbnd1lem3 27611 pntpbnd1a 27725

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