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Theorem logdiflbnd 25133

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 12119	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		rpge0 12126	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 𝐴)
3	1, 2	ge0p1rpd 12185	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ∈ ℝ⁺)
4	3	rprecred 12166	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 + 1)) ∈ ℝ)
5		1red 10356	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℝ)
6		0le1 10874	. . . . . . 7 ⊢ 0 ≤ 1
7	6	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 1)
8	5, 3, 7	divge0d 12195	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ (1 / (𝐴 + 1)))
9		id 22	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ⁺)
10	5, 9	ltaddrp2d 12189	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 1 < (𝐴 + 1))
11	1, 5	readdcld 10385	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ∈ ℝ)
12	11	recnd 10384	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ∈ ℂ)
13	12	mulid1d 10373	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) · 1) = (𝐴 + 1))
14	10, 13	breqtrrd 4900	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 1 < ((𝐴 + 1) · 1))
15	5, 5, 3	ltdivmuld 12206	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((1 / (𝐴 + 1)) < 1 ↔ 1 < ((𝐴 + 1) · 1)))
16	14, 15	mpbird 249	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 + 1)) < 1)
17	4, 8, 16	eflegeo 15222	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / (𝐴 + 1))) ≤ (1 / (1 − (1 / (𝐴 + 1)))))
18	5	recnd 10384	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℂ)
19	3	rpne0d 12160	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 + 1) ≠ 0)
20	12, 18, 12, 19	divsubdird 11165	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 + 1) − 1) / (𝐴 + 1)) = (((𝐴 + 1) / (𝐴 + 1)) − (1 / (𝐴 + 1))))
21	1	recnd 10384	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
22	21, 18	pncand 10713	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) − 1) = 𝐴)
23	22	oveq1d 6919	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 + 1) − 1) / (𝐴 + 1)) = (𝐴 / (𝐴 + 1)))
24	12, 19	dividd 11124	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) / (𝐴 + 1)) = 1)
25	24	oveq1d 6919	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (((𝐴 + 1) / (𝐴 + 1)) − (1 / (𝐴 + 1))) = (1 − (1 / (𝐴 + 1))))
26	20, 23, 25	3eqtr3rd 2869	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 − (1 / (𝐴 + 1))) = (𝐴 / (𝐴 + 1)))
27	26	oveq2d 6920	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (1 − (1 / (𝐴 + 1)))) = (1 / (𝐴 / (𝐴 + 1))))
28		rpne0 12129	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
29	21, 12, 28, 19	recdivd 11143	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 / (𝐴 + 1))) = ((𝐴 + 1) / 𝐴))
30	27, 29	eqtrd 2860	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (1 − (1 / (𝐴 + 1)))) = ((𝐴 + 1) / 𝐴))
31	17, 30	breqtrd 4898	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / (𝐴 + 1))) ≤ ((𝐴 + 1) / 𝐴))
32	4	rpefcld 15206	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / (𝐴 + 1))) ∈ ℝ⁺)
33	3, 9	rpdivcld 12172	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((𝐴 + 1) / 𝐴) ∈ ℝ⁺)
34	32, 33	logled 24771	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((exp‘(1 / (𝐴 + 1))) ≤ ((𝐴 + 1) / 𝐴) ↔ (log‘(exp‘(1 / (𝐴 + 1)))) ≤ (log‘((𝐴 + 1) / 𝐴))))
35	31, 34	mpbid 224	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / (𝐴 + 1)))) ≤ (log‘((𝐴 + 1) / 𝐴)))
36	4	relogefd 24772	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / (𝐴 + 1)))) = (1 / (𝐴 + 1)))
37	3, 9	relogdivd 24770	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((𝐴 + 1) / 𝐴)) = ((log‘(𝐴 + 1)) − (log‘𝐴)))
38	35, 36, 37	3brtr3d 4903	1 ⊢ (𝐴 ∈ ℝ⁺ → (1 / (𝐴 + 1)) ≤ ((log‘(𝐴 + 1)) − (log‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2166 class class class wbr 4872 ‘cfv 6122 (class class class)co 6904 0cc0 10251 1c1 10252 + caddc 10254 · cmul 10256 < clt 10390 ≤ cle 10391 − cmin 10584 / cdiv 11008 ℝ⁺crp 12111 expce 15163 logclog 24699

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-inf2 8814 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-pre-sup 10329 ax-addf 10330 ax-mulf 10331

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-supp 7559 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-2o 7826 df-oadd 7829 df-er 8008 df-map 8123 df-pm 8124 df-ixp 8175 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fsupp 8544 df-fi 8585 df-sup 8616 df-inf 8617 df-oi 8683 df-card 9077 df-cda 9304 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-div 11009 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-q 12071 df-rp 12112 df-xneg 12231 df-xadd 12232 df-xmul 12233 df-ioo 12466 df-ioc 12467 df-ico 12468 df-icc 12469 df-fz 12619 df-fzo 12760 df-fl 12887 df-mod 12963 df-seq 13095 df-exp 13154 df-fac 13353 df-bc 13382 df-hash 13410 df-shft 14183 df-cj 14215 df-re 14216 df-im 14217 df-sqrt 14351 df-abs 14352 df-limsup 14578 df-clim 14595 df-rlim 14596 df-sum 14793 df-ef 15169 df-sin 15171 df-cos 15172 df-pi 15174 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-starv 16319 df-sca 16320 df-vsca 16321 df-ip 16322 df-tset 16323 df-ple 16324 df-ds 16326 df-unif 16327 df-hom 16328 df-cco 16329 df-rest 16435 df-topn 16436 df-0g 16454 df-gsum 16455 df-topgen 16456 df-pt 16457 df-prds 16460 df-xrs 16514 df-qtop 16519 df-imas 16520 df-xps 16522 df-mre 16598 df-mrc 16599 df-acs 16601 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-mulg 17894 df-cntz 18099 df-cmn 18547 df-psmet 20097 df-xmet 20098 df-met 20099 df-bl 20100 df-mopn 20101 df-fbas 20102 df-fg 20103 df-cnfld 20106 df-top 21068 df-topon 21085 df-topsp 21107 df-bases 21120 df-cld 21193 df-ntr 21194 df-cls 21195 df-nei 21272 df-lp 21310 df-perf 21311 df-cn 21401 df-cnp 21402 df-haus 21489 df-tx 21735 df-hmeo 21928 df-fil 22019 df-fm 22111 df-flim 22112 df-flf 22113 df-xms 22494 df-ms 22495 df-tms 22496 df-cncf 23050 df-limc 24028 df-dv 24029 df-log 24701

This theorem is referenced by: lgamgulmlem3 25169

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