logdivlt - Metamath Proof Explorer

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Theorem logdivlt 26585

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 26584	. . . . . 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 1119	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ e ≤ 𝐴) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
4	3	an32s 653	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
5	4	adantrr 718	. 2 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
6		fveq2 6841	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (log‘𝐴) = (log‘𝐵))
7		id 22	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
8	6, 7	oveq12d 7385	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((log‘𝐴) / 𝐴) = ((log‘𝐵) / 𝐵))
9	8	eqcomd 2743	. . . . . 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 26584	. . . . . . . . . 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 1119	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ e ≤ 𝐵) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
14	13	an32s 653	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ ∧ e ≤ 𝐵) ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
15	14	adantrr 718	. . . . . 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 967	. . . 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 16174	. . . . . . . 8 ⊢ 0 < e
21		0re 11146	. . . . . . . . 9 ⊢ 0 ∈ ℝ
22		ere 16054	. . . . . . . . 9 ⊢ e ∈ ℝ
23		ltletr 11238	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ e ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < e ∧ e ≤ 𝐵) → 0 < 𝐵))
24	21, 22, 23	mp3an12 1454	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → ((0 < e ∧ e ≤ 𝐵) → 0 < 𝐵))
25	20, 24	mpani 697	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (e ≤ 𝐵 → 0 < 𝐵))
26	25	imp 406	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 0 < 𝐵)
27	19, 26	elrpd 12983	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 𝐵 ∈ ℝ⁺)
28		relogcl 26539	. . . . . 6 ⊢ (𝐵 ∈ ℝ⁺ → (log‘𝐵) ∈ ℝ)
29		rerpdivcl 12974	. . . . . 6 ⊢ (((log‘𝐵) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((log‘𝐵) / 𝐵) ∈ ℝ)
30	28, 29	mpancom 689	. . . . 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 11238	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ e ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < e ∧ e ≤ 𝐴) → 0 < 𝐴))
34	21, 22, 33	mp3an12 1454	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((0 < e ∧ e ≤ 𝐴) → 0 < 𝐴))
35	20, 34	mpani 697	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (e ≤ 𝐴 → 0 < 𝐴))
36	35	imp 406	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 0 < 𝐴)
37	32, 36	elrpd 12983	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
38		relogcl 26539	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
39		rerpdivcl 12974	. . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → ((log‘𝐴) / 𝐴) ∈ ℝ)
40	38, 39	mpancom 689	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) / 𝐴) ∈ ℝ)
41	37, 40	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → ((log‘𝐴) / 𝐴) ∈ ℝ)
42		axlttri 11217	. . . 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 598	. . 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 11217	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
45	44	ad2ant2r 748	. . 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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5086 ‘cfv 6499 (class class class)co 7367 ℝcr 11037 0cc0 11038 < clt 11179 ≤ cle 11180 / cdiv 11807 ℝ⁺crp 12942 eceu 16027 logclog 26518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15649 df-ef 16032 df-e 16033 df-sin 16034 df-cos 16035 df-pi 16037 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520

This theorem is referenced by: logdivle 26586 bposlem7 27253 chebbnd1lem2 27433 chebbnd1lem3 27434 pntpbnd1a 27548

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