rege1logbrege0 - Mathbox for Alexander van der Vekens

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Theorem rege1logbrege0 48538

Description: The general logarithm, with a real base greater than 1, for a real number greater than or equal to 1 is greater than or equal to 0. (Contributed by AV, 25-May-2020.)

**Assertion**
Ref	Expression
rege1logbrege0	⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 log_b 𝑋))

**Proof of Theorem rege1logbrege0**
Step	Hyp	Ref	Expression
1		1re 11235	. . . . . . 7 ⊢ 1 ∈ ℝ
2		elicopnf 13462	. . . . . . 7 ⊢ (1 ∈ ℝ → (𝑋 ∈ (1[,)+∞) ↔ (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋)))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ (𝑋 ∈ (1[,)+∞) ↔ (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋))
4		id 22	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋))
5	3, 4	sylbi 217	. . . . 5 ⊢ (𝑋 ∈ (1[,)+∞) → (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋))
6	5	adantl 481	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋))
7		logge0 26566	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 0 ≤ (log‘𝑋))
8	6, 7	syl 17	. . 3 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (log‘𝑋))
9		simpl 482	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℝ)
10		0lt1 11759	. . . . . . . . . 10 ⊢ 0 < 1
11		0red 11238	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 0 ∈ ℝ)
12		1red 11236	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 1 ∈ ℝ)
13		id 22	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 𝑋 ∈ ℝ)
14		ltletr 11327	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑋 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝑋) → 0 < 𝑋))
15	11, 12, 13, 14	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝑋) → 0 < 𝑋))
16	10, 15	mpani 696	. . . . . . . . 9 ⊢ (𝑋 ∈ ℝ → (1 ≤ 𝑋 → 0 < 𝑋))
17	16	imp 406	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 0 < 𝑋)
18	9, 17	elrpd 13048	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℝ⁺)
19	3, 18	sylbi 217	. . . . . 6 ⊢ (𝑋 ∈ (1[,)+∞) → 𝑋 ∈ ℝ⁺)
20	19	relogcld 26584	. . . . 5 ⊢ (𝑋 ∈ (1[,)+∞) → (log‘𝑋) ∈ ℝ)
21	20	adantl 481	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (log‘𝑋) ∈ ℝ)
22		1xr 11294	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
23		elioopnf 13460	. . . . . . . 8 ⊢ (1 ∈ ℝ^* → (𝐵 ∈ (1(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 1 < 𝐵)))
24	22, 23	ax-mp 5	. . . . . . 7 ⊢ (𝐵 ∈ (1(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 1 < 𝐵))
25		simpl 482	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℝ)
26		0red 11238	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℝ)
27		1red 11236	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 1 ∈ ℝ)
28		id 22	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ)
29		lttr 11311	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝐵) → 0 < 𝐵))
30	26, 27, 28, 29	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → ((0 < 1 ∧ 1 < 𝐵) → 0 < 𝐵))
31	10, 30	mpani 696	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (1 < 𝐵 → 0 < 𝐵))
32	31	imp 406	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 0 < 𝐵)
33	25, 32	elrpd 13048	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℝ⁺)
34	24, 33	sylbi 217	. . . . . 6 ⊢ (𝐵 ∈ (1(,)+∞) → 𝐵 ∈ ℝ⁺)
35	34	relogcld 26584	. . . . 5 ⊢ (𝐵 ∈ (1(,)+∞) → (log‘𝐵) ∈ ℝ)
36	35	adantr 480	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (log‘𝐵) ∈ ℝ)
37		regt1loggt0 48516	. . . . 5 ⊢ (𝐵 ∈ (1(,)+∞) → 0 < (log‘𝐵))
38	37	adantr 480	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 < (log‘𝐵))
39		ge0div 12109	. . . 4 ⊢ (((log‘𝑋) ∈ ℝ ∧ (log‘𝐵) ∈ ℝ ∧ 0 < (log‘𝐵)) → (0 ≤ (log‘𝑋) ↔ 0 ≤ ((log‘𝑋) / (log‘𝐵))))
40	21, 36, 38, 39	syl3anc 1373	. . 3 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (0 ≤ (log‘𝑋) ↔ 0 ≤ ((log‘𝑋) / (log‘𝐵))))
41	8, 40	mpbid 232	. 2 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ ((log‘𝑋) / (log‘𝐵)))
42		recn 11219	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
43	42	adantr 480	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℂ)
44	32	gt0ne0d 11801	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ≠ 0)
45	27, 28	ltlend 11380	. . . . . 6 ⊢ (𝐵 ∈ ℝ → (1 < 𝐵 ↔ (1 ≤ 𝐵 ∧ 𝐵 ≠ 1)))
46	45	simplbda 499	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ≠ 1)
47	43, 44, 46	3jca 1128	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
48		eldifpr 4634	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
49	47, 24, 48	3imtr4i 292	. . 3 ⊢ (𝐵 ∈ (1(,)+∞) → 𝐵 ∈ (ℂ ∖ {0, 1}))
50		recn 11219	. . . . . 6 ⊢ (𝑋 ∈ ℝ → 𝑋 ∈ ℂ)
51	50	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℂ)
52	17	gt0ne0d 11801	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ≠ 0)
53	51, 52	jca 511	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
54		eldifsn 4762	. . . 4 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
55	53, 3, 54	3imtr4i 292	. . 3 ⊢ (𝑋 ∈ (1[,)+∞) → 𝑋 ∈ (ℂ ∖ {0}))
56		logbval 26728	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝑋) = ((log‘𝑋) / (log‘𝐵)))
57	49, 55, 56	syl2an 596	. 2 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (𝐵 log_b 𝑋) = ((log‘𝑋) / (log‘𝐵)))
58	41, 57	breqtrrd 5147	1 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 log_b 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∖ cdif 3923 {csn 4601 {cpr 4603 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 +∞cpnf 11266 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 / cdiv 11894 ℝ⁺crp 13008 (,)cioo 13362 [,)cico 13364 logclog 26515 log_b clogb 26726

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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-pi 16088 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-limc 25819 df-dv 25820 df-log 26517 df-logb 26727

This theorem is referenced by: rege1logbzge0 48539

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