rege1logbrege0 - Mathbox for Alexander van der Vekens

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Theorem rege1logbrege0 48547

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 11174	. . . . . . 7 ⊢ 1 ∈ ℝ
2		elicopnf 13406	. . . . . . 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 26514	. . . 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 11700	. . . . . . . . . 10 ⊢ 0 < 1
11		0red 11177	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 0 ∈ ℝ)
12		1red 11175	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 1 ∈ ℝ)
13		id 22	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 𝑋 ∈ ℝ)
14		ltletr 11266	. . . . . . . . . . 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 12992	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℝ⁺)
19	3, 18	sylbi 217	. . . . . 6 ⊢ (𝑋 ∈ (1[,)+∞) → 𝑋 ∈ ℝ⁺)
20	19	relogcld 26532	. . . . 5 ⊢ (𝑋 ∈ (1[,)+∞) → (log‘𝑋) ∈ ℝ)
21	20	adantl 481	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (log‘𝑋) ∈ ℝ)
22		1xr 11233	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
23		elioopnf 13404	. . . . . . . 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 11177	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℝ)
27		1red 11175	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 1 ∈ ℝ)
28		id 22	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ)
29		lttr 11250	. . . . . . . . . . 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 12992	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℝ⁺)
34	24, 33	sylbi 217	. . . . . 6 ⊢ (𝐵 ∈ (1(,)+∞) → 𝐵 ∈ ℝ⁺)
35	34	relogcld 26532	. . . . 5 ⊢ (𝐵 ∈ (1(,)+∞) → (log‘𝐵) ∈ ℝ)
36	35	adantr 480	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (log‘𝐵) ∈ ℝ)
37		regt1loggt0 48525	. . . . 5 ⊢ (𝐵 ∈ (1(,)+∞) → 0 < (log‘𝐵))
38	37	adantr 480	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 < (log‘𝐵))
39		ge0div 12050	. . . 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 11158	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
43	42	adantr 480	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℂ)
44	32	gt0ne0d 11742	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ≠ 0)
45	27, 28	ltlend 11319	. . . . . 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 4622	. . . 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 11158	. . . . . 6 ⊢ (𝑋 ∈ ℝ → 𝑋 ∈ ℂ)
51	50	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℂ)
52	17	gt0ne0d 11742	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ≠ 0)
53	51, 52	jca 511	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
54		eldifsn 4750	. . . 4 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
55	53, 3, 54	3imtr4i 292	. . 3 ⊢ (𝑋 ∈ (1[,)+∞) → 𝑋 ∈ (ℂ ∖ {0}))
56		logbval 26676	. . 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 5135	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 2109 ≠ wne 2925 ∖ cdif 3911 {csn 4589 {cpr 4591 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 / cdiv 11835 ℝ⁺crp 12951 (,)cioo 13306 [,)cico 13308 logclog 26463 log_b clogb 26674

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-sin 16035 df-cos 16036 df-pi 16038 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-log 26465 df-logb 26675

This theorem is referenced by: rege1logbzge0 48548

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