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Theorem logdiflbnd 24941

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 12052	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		rpge0 12058	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 𝐴)
3	1, 2	ge0p1rpd 12115	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ∈ ℝ⁺)
4	3	rprecred 12096	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 + 1)) ∈ ℝ)
5		1red 10267	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℝ)
6		0le1 10763	. . . . . . 7 ⊢ 0 ≤ 1
7	6	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 1)
8	5, 3, 7	divge0d 12125	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ (1 / (𝐴 + 1)))
9		id 22	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ⁺)
10	5, 9	ltaddrp2d 12119	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 1 < (𝐴 + 1))
11	1, 5	readdcld 10281	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ∈ ℝ)
12	11	recnd 10280	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ∈ ℂ)
13	12	mulid1d 10269	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) · 1) = (𝐴 + 1))
14	10, 13	breqtrrd 4832	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 1 < ((𝐴 + 1) · 1))
15	5, 5, 3	ltdivmuld 12136	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((1 / (𝐴 + 1)) < 1 ↔ 1 < ((𝐴 + 1) · 1)))
16	14, 15	mpbird 247	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 + 1)) < 1)
17	4, 8, 16	eflegeo 15070	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / (𝐴 + 1))) ≤ (1 / (1 − (1 / (𝐴 + 1)))))
18	5	recnd 10280	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℂ)
19	3	rpne0d 12090	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ≠ 0)
20	12, 18, 12, 19	divsubdird 11052	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 + 1) − 1) / (𝐴 + 1)) = (((𝐴 + 1) / (𝐴 + 1)) − (1 / (𝐴 + 1))))
21	1	recnd 10280	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
22	21, 18	pncand 10605	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) − 1) = 𝐴)
23	22	oveq1d 6829	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 + 1) − 1) / (𝐴 + 1)) = (𝐴 / (𝐴 + 1)))
24	12, 19	dividd 11011	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) / (𝐴 + 1)) = 1)
25	24	oveq1d 6829	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 + 1) / (𝐴 + 1)) − (1 / (𝐴 + 1))) = (1 − (1 / (𝐴 + 1))))
26	20, 23, 25	3eqtr3rd 2803	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (1 / (𝐴 + 1))) = (𝐴 / (𝐴 + 1)))
27	26	oveq2d 6830	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (1 − (1 / (𝐴 + 1)))) = (1 / (𝐴 / (𝐴 + 1))))
28		rpne0 12061	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
29	21, 12, 28, 19	recdivd 11030	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 / (𝐴 + 1))) = ((𝐴 + 1) / 𝐴))
30	27, 29	eqtrd 2794	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (1 − (1 / (𝐴 + 1)))) = ((𝐴 + 1) / 𝐴))
31	17, 30	breqtrd 4830	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / (𝐴 + 1))) ≤ ((𝐴 + 1) / 𝐴))
32	4	rpefcld 15054	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / (𝐴 + 1))) ∈ ℝ⁺)
33	3, 9	rpdivcld 12102	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) / 𝐴) ∈ ℝ⁺)
34	32, 33	logled 24593	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((exp‘(1 / (𝐴 + 1))) ≤ ((𝐴 + 1) / 𝐴) ↔ (log‘(exp‘(1 / (𝐴 + 1)))) ≤ (log‘((𝐴 + 1) / 𝐴))))
35	31, 34	mpbid 222	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / (𝐴 + 1)))) ≤ (log‘((𝐴 + 1) / 𝐴)))
36	4	relogefd 24594	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / (𝐴 + 1)))) = (1 / (𝐴 + 1)))
37	3, 9	relogdivd 24592	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((𝐴 + 1) / 𝐴)) = ((log‘(𝐴 + 1)) − (log‘𝐴)))
38	35, 36, 37	3brtr3d 4835	1 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6814 0cc0 10148 1c1 10149 + caddc 10151 · cmul 10153 < clt 10286 ≤ cle 10287 − cmin 10478 / cdiv 10896 ℝ⁺crp 12045 expce 15011 logclog 24521

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 ax-addf 10227 ax-mulf 10228

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-supp 7465 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-er 7913 df-map 8027 df-pm 8028 df-ixp 8077 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-fsupp 8443 df-fi 8484 df-sup 8515 df-inf 8516 df-oi 8582 df-card 8975 df-cda 9202 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-q 12002 df-rp 12046 df-xneg 12159 df-xadd 12160 df-xmul 12161 df-ioo 12392 df-ioc 12393 df-ico 12394 df-icc 12395 df-fz 12540 df-fzo 12680 df-fl 12807 df-mod 12883 df-seq 13016 df-exp 13075 df-fac 13275 df-bc 13304 df-hash 13332 df-shft 14026 df-cj 14058 df-re 14059 df-im 14060 df-sqrt 14194 df-abs 14195 df-limsup 14421 df-clim 14438 df-rlim 14439 df-sum 14636 df-ef 15017 df-sin 15019 df-cos 15020 df-pi 15022 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-hom 16188 df-cco 16189 df-rest 16305 df-topn 16306 df-0g 16324 df-gsum 16325 df-topgen 16326 df-pt 16327 df-prds 16330 df-xrs 16384 df-qtop 16389 df-imas 16390 df-xps 16392 df-mre 16468 df-mrc 16469 df-acs 16471 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-submnd 17557 df-mulg 17762 df-cntz 17970 df-cmn 18415 df-psmet 19960 df-xmet 19961 df-met 19962 df-bl 19963 df-mopn 19964 df-fbas 19965 df-fg 19966 df-cnfld 19969 df-top 20921 df-topon 20938 df-topsp 20959 df-bases 20972 df-cld 21045 df-ntr 21046 df-cls 21047 df-nei 21124 df-lp 21162 df-perf 21163 df-cn 21253 df-cnp 21254 df-haus 21341 df-tx 21587 df-hmeo 21780 df-fil 21871 df-fm 21963 df-flim 21964 df-flf 21965 df-xms 22346 df-ms 22347 df-tms 22348 df-cncf 22902 df-limc 23849 df-dv 23850 df-log 24523

This theorem is referenced by: lgamgulmlem3 24977

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