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Theorem logdiflbnd 26189

Description: Lower bound on the difference of logs. (Contributed by Mario Carneiro, 3-Jul-2017.)

**Assertion**
Ref	Expression
logdiflbnd	⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴)))

**Proof of Theorem logdiflbnd**
Step	Hyp	Ref	Expression
1		rpre 12784	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		rpge0 12789	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 𝐴)
3	1, 2	ge0p1rpd 12848	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ∈ ℝ⁺)
4	3	rprecred 12829	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 + 1)) ∈ ℝ)
5		1red 11022	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℝ)
6		0le1 11544	. . . . . . 7 ⊢ 0 ≤ 1
7	6	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 1)
8	5, 3, 7	divge0d 12858	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ (1 / (𝐴 + 1)))
9		id 22	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ⁺)
10	5, 9	ltaddrp2d 12852	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 1 < (𝐴 + 1))
11	1, 5	readdcld 11050	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ∈ ℝ)
12	11	recnd 11049	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ∈ ℂ)
13	12	mulid1d 11038	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) · 1) = (𝐴 + 1))
14	10, 13	breqtrrd 5109	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 1 < ((𝐴 + 1) · 1))
15	5, 5, 3	ltdivmuld 12869	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((1 / (𝐴 + 1)) < 1 ↔ 1 < ((𝐴 + 1) · 1)))
16	14, 15	mpbird 257	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 + 1)) < 1)
17	4, 8, 16	eflegeo 15875	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / (𝐴 + 1))) ≤ (1 / (1 − (1 / (𝐴 + 1)))))
18	5	recnd 11049	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℂ)
19	3	rpne0d 12823	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ≠ 0)
20	12, 18, 12, 19	divsubdird 11836	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 + 1) − 1) / (𝐴 + 1)) = (((𝐴 + 1) / (𝐴 + 1)) − (1 / (𝐴 + 1))))
21	1	recnd 11049	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
22	21, 18	pncand 11379	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) − 1) = 𝐴)
23	22	oveq1d 7322	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 + 1) − 1) / (𝐴 + 1)) = (𝐴 / (𝐴 + 1)))
24	12, 19	dividd 11795	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) / (𝐴 + 1)) = 1)
25	24	oveq1d 7322	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 + 1) / (𝐴 + 1)) − (1 / (𝐴 + 1))) = (1 − (1 / (𝐴 + 1))))
26	20, 23, 25	3eqtr3rd 2785	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (1 / (𝐴 + 1))) = (𝐴 / (𝐴 + 1)))
27	26	oveq2d 7323	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (1 − (1 / (𝐴 + 1)))) = (1 / (𝐴 / (𝐴 + 1))))
28		rpne0 12792	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
29	21, 12, 28, 19	recdivd 11814	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 / (𝐴 + 1))) = ((𝐴 + 1) / 𝐴))
30	27, 29	eqtrd 2776	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (1 − (1 / (𝐴 + 1)))) = ((𝐴 + 1) / 𝐴))
31	17, 30	breqtrd 5107	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / (𝐴 + 1))) ≤ ((𝐴 + 1) / 𝐴))
32	4	rpefcld 15859	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / (𝐴 + 1))) ∈ ℝ⁺)
33	3, 9	rpdivcld 12835	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) / 𝐴) ∈ ℝ⁺)
34	32, 33	logled 25827	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((exp‘(1 / (𝐴 + 1))) ≤ ((𝐴 + 1) / 𝐴) ↔ (log‘(exp‘(1 / (𝐴 + 1)))) ≤ (log‘((𝐴 + 1) / 𝐴))))
35	31, 34	mpbid 231	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / (𝐴 + 1)))) ≤ (log‘((𝐴 + 1) / 𝐴)))
36	4	relogefd 25828	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / (𝐴 + 1)))) = (1 / (𝐴 + 1)))
37	3, 9	relogdivd 25826	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((𝐴 + 1) / 𝐴)) = ((log‘(𝐴 + 1)) − (log‘𝐴)))
38	35, 36, 37	3brtr3d 5112	1 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2104 class class class wbr 5081 ‘cfv 6458 (class class class)co 7307 0cc0 10917 1c1 10918 + caddc 10920 · cmul 10922 < clt 11055 ≤ cle 11056 − cmin 11251 / cdiv 11678 ℝ⁺crp 12776 expce 15816 logclog 25755

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-inf2 9443 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 ax-pre-sup 10995 ax-addf 10996 ax-mulf 10997

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-iin 4934 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-of 7565 df-om 7745 df-1st 7863 df-2nd 7864 df-supp 8009 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-2o 8329 df-er 8529 df-map 8648 df-pm 8649 df-ixp 8717 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-fsupp 9173 df-fi 9214 df-sup 9245 df-inf 9246 df-oi 9313 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-div 11679 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-5 12085 df-6 12086 df-7 12087 df-8 12088 df-9 12089 df-n0 12280 df-z 12366 df-dec 12484 df-uz 12629 df-q 12735 df-rp 12777 df-xneg 12894 df-xadd 12895 df-xmul 12896 df-ioo 13129 df-ioc 13130 df-ico 13131 df-icc 13132 df-fz 13286 df-fzo 13429 df-fl 13558 df-mod 13636 df-seq 13768 df-exp 13829 df-fac 14034 df-bc 14063 df-hash 14091 df-shft 14823 df-cj 14855 df-re 14856 df-im 14857 df-sqrt 14991 df-abs 14992 df-limsup 15225 df-clim 15242 df-rlim 15243 df-sum 15443 df-ef 15822 df-sin 15824 df-cos 15825 df-pi 15827 df-struct 16893 df-sets 16910 df-slot 16928 df-ndx 16940 df-base 16958 df-ress 16987 df-plusg 17020 df-mulr 17021 df-starv 17022 df-sca 17023 df-vsca 17024 df-ip 17025 df-tset 17026 df-ple 17027 df-ds 17029 df-unif 17030 df-hom 17031 df-cco 17032 df-rest 17178 df-topn 17179 df-0g 17197 df-gsum 17198 df-topgen 17199 df-pt 17200 df-prds 17203 df-xrs 17258 df-qtop 17263 df-imas 17264 df-xps 17266 df-mre 17340 df-mrc 17341 df-acs 17343 df-mgm 18371 df-sgrp 18420 df-mnd 18431 df-submnd 18476 df-mulg 18746 df-cntz 18968 df-cmn 19433 df-psmet 20634 df-xmet 20635 df-met 20636 df-bl 20637 df-mopn 20638 df-fbas 20639 df-fg 20640 df-cnfld 20643 df-top 22088 df-topon 22105 df-topsp 22127 df-bases 22141 df-cld 22215 df-ntr 22216 df-cls 22217 df-nei 22294 df-lp 22332 df-perf 22333 df-cn 22423 df-cnp 22424 df-haus 22511 df-tx 22758 df-hmeo 22951 df-fil 23042 df-fm 23134 df-flim 23135 df-flf 23136 df-xms 23518 df-ms 23519 df-tms 23520 df-cncf 24086 df-limc 25075 df-dv 25076 df-log 25757

This theorem is referenced by: lgamgulmlem3 26225

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