logdivlt - Metamath Proof Explorer

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Theorem logdivlt 25204

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 25203	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐴 < 𝐵) → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))
2	1	ex 415	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
3	2	3expa 1114	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ e ≤ 𝐴) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
4	3	an32s 650	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
5	4	adantrr 715	. 2 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
6		fveq2 6670	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (log‘𝐴) = (log‘𝐵))
7		id 22	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵)
8	6, 7	oveq12d 7174	. . . . . . 7 ⊢ (𝐴 = 𝐵 → ((log‘𝐴) / 𝐴) = ((log‘𝐵) / 𝐵))
9	8	eqcomd 2827	. . . . . 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 25203	. . . . . . . . . 10 ⊢ (((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ e ≤ 𝐵) ∧ 𝐵 < 𝐴) → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵))
12	11	ex 415	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ e ≤ 𝐵) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
13	12	3expa 1114	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ e ≤ 𝐵) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
14	13	an32s 650	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ ∧ e ≤ 𝐵) ∧ 𝐴 ∈ ℝ) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
15	14	adantrr 715	. . . . . 6 ⊢ (((𝐵 ∈ ℝ ∧ e ≤ 𝐵) ∧ (𝐴 ∈ ℝ ∧ e ≤ 𝐴)) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
16	15	ancoms 461	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐵 < 𝐴 → ((log‘𝐴) / 𝐴) < ((log‘𝐵) / 𝐵)))
17	10, 16	orim12d 961	. . . 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 155	. . 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 485	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 𝐵 ∈ ℝ)
20		epos 15560	. . . . . . . 8 ⊢ 0 < e
21		0re 10643	. . . . . . . . 9 ⊢ 0 ∈ ℝ
22		ere 15442	. . . . . . . . 9 ⊢ e ∈ ℝ
23		ltletr 10732	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ e ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < e ∧ e ≤ 𝐵) → 0 < 𝐵))
24	21, 22, 23	mp3an12 1447	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → ((0 < e ∧ e ≤ 𝐵) → 0 < 𝐵))
25	20, 24	mpani 694	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → (e ≤ 𝐵 → 0 < 𝐵))
26	25	imp 409	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 0 < 𝐵)
27	19, 26	elrpd 12429	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → 𝐵 ∈ ℝ⁺)
28		relogcl 25159	. . . . . 6 ⊢ (𝐵 ∈ ℝ⁺ → (log‘𝐵) ∈ ℝ)
29		rerpdivcl 12420	. . . . . 6 ⊢ (((log‘𝐵) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((log‘𝐵) / 𝐵) ∈ ℝ)
30	28, 29	mpancom 686	. . . . 5 ⊢ (𝐵 ∈ ℝ⁺ → ((log‘𝐵) / 𝐵) ∈ ℝ)
31	27, 30	syl 17	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ e ≤ 𝐵) → ((log‘𝐵) / 𝐵) ∈ ℝ)
32		simpl 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 𝐴 ∈ ℝ)
33		ltletr 10732	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ e ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((0 < e ∧ e ≤ 𝐴) → 0 < 𝐴))
34	21, 22, 33	mp3an12 1447	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((0 < e ∧ e ≤ 𝐴) → 0 < 𝐴))
35	20, 34	mpani 694	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (e ≤ 𝐴 → 0 < 𝐴))
36	35	imp 409	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 0 < 𝐴)
37	32, 36	elrpd 12429	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → 𝐴 ∈ ℝ⁺)
38		relogcl 25159	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
39		rerpdivcl 12420	. . . . . 6 ⊢ (((log‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → ((log‘𝐴) / 𝐴) ∈ ℝ)
40	38, 39	mpancom 686	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) / 𝐴) ∈ ℝ)
41	37, 40	syl 17	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ e ≤ 𝐴) → ((log‘𝐴) / 𝐴) ∈ ℝ)
42		axlttri 10712	. . . 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 10712	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
45	44	ad2ant2r 745	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
46	18, 43, 45	3imtr4d 296	. 2 ⊢ (((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴) → 𝐴 < 𝐵))
47	5, 46	impbid 214	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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 < clt 10675 ≤ cle 10676 / cdiv 11297 ℝ⁺crp 12390 eceu 15416 logclog 25138

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-e 15422 df-sin 15423 df-cos 15424 df-pi 15426 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140

This theorem is referenced by: logdivle 25205 bposlem7 25866 chebbnd1lem2 26046 chebbnd1lem3 26047 pntpbnd1a 26161

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