rege1logbrege0 - Mathbox for Alexander van der Vekens

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Theorem rege1logbrege0 42779

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 10152	. . . . . . 7 ⊢ 1 ∈ ℝ
2		elicopnf 12383	. . . . . . 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 207	. . . . 5 ⊢ (𝑋 ∈ (1[,)+∞) → (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋))
6	5	adantl 473	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (𝑋 ∈ ℝ ∧ 1 ≤ 𝑋))
7		logge0 24471	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 0 ≤ (log‘𝑋))
8	6, 7	syl 17	. . 3 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (log‘𝑋))
9		simpl 474	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℝ)
10		0lt1 10663	. . . . . . . . . 10 ⊢ 0 < 1
11		0red 10154	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 0 ∈ ℝ)
12		1red 10168	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 1 ∈ ℝ)
13		id 22	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 𝑋 ∈ ℝ)
14		ltletr 10242	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑋 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝑋) → 0 < 𝑋))
15	11, 12, 13, 14	syl3anc 1439	. . . . . . . . . 10 ⊢ (𝑋 ∈ ℝ → ((0 < 1 ∧ 1 ≤ 𝑋) → 0 < 𝑋))
16	10, 15	mpani 714	. . . . . . . . 9 ⊢ (𝑋 ∈ ℝ → (1 ≤ 𝑋 → 0 < 𝑋))
17	16	imp 444	. . . . . . . 8 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 0 < 𝑋)
18	9, 17	elrpd 11983	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℝ⁺)
19	3, 18	sylbi 207	. . . . . 6 ⊢ (𝑋 ∈ (1[,)+∞) → 𝑋 ∈ ℝ⁺)
20	19	relogcld 24489	. . . . 5 ⊢ (𝑋 ∈ (1[,)+∞) → (log‘𝑋) ∈ ℝ)
21	20	adantl 473	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (log‘𝑋) ∈ ℝ)
22	1	rexri 10210	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
23		elioopnf 12381	. . . . . . . 8 ⊢ (1 ∈ ℝ^* → (𝐵 ∈ (1(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 1 < 𝐵)))
24	22, 23	ax-mp 5	. . . . . . 7 ⊢ (𝐵 ∈ (1(,)+∞) ↔ (𝐵 ∈ ℝ ∧ 1 < 𝐵))
25		simpl 474	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℝ)
26		0red 10154	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℝ)
27		1red 10168	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 1 ∈ ℝ)
28		id 22	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ)
29		lttr 10227	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝐵) → 0 < 𝐵))
30	26, 27, 28, 29	syl3anc 1439	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → ((0 < 1 ∧ 1 < 𝐵) → 0 < 𝐵))
31	10, 30	mpani 714	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (1 < 𝐵 → 0 < 𝐵))
32	31	imp 444	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 0 < 𝐵)
33	25, 32	elrpd 11983	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℝ⁺)
34	24, 33	sylbi 207	. . . . . 6 ⊢ (𝐵 ∈ (1(,)+∞) → 𝐵 ∈ ℝ⁺)
35	34	relogcld 24489	. . . . 5 ⊢ (𝐵 ∈ (1(,)+∞) → (log‘𝐵) ∈ ℝ)
36	35	adantr 472	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (log‘𝐵) ∈ ℝ)
37		regt1loggt0 42757	. . . . 5 ⊢ (𝐵 ∈ (1(,)+∞) → 0 < (log‘𝐵))
38	37	adantr 472	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 < (log‘𝐵))
39		ge0div 11003	. . . 4 ⊢ (((log‘𝑋) ∈ ℝ ∧ (log‘𝐵) ∈ ℝ ∧ 0 < (log‘𝐵)) → (0 ≤ (log‘𝑋) ↔ 0 ≤ ((log‘𝑋) / (log‘𝐵))))
40	21, 36, 38, 39	syl3anc 1439	. . 3 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (0 ≤ (log‘𝑋) ↔ 0 ≤ ((log‘𝑋) / (log‘𝐵))))
41	8, 40	mpbid 222	. 2 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ ((log‘𝑋) / (log‘𝐵)))
42		recn 10139	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
43	42	adantr 472	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℂ)
44	32	gt0ne0d 10705	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ≠ 0)
45	27, 28	ltlend 10295	. . . . . 6 ⊢ (𝐵 ∈ ℝ → (1 < 𝐵 ↔ (1 ≤ 𝐵 ∧ 𝐵 ≠ 1)))
46	45	simplbda 655	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ≠ 1)
47	43, 44, 46	3jca 1379	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
48		eldifpr 4312	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
49	47, 24, 48	3imtr4i 281	. . 3 ⊢ (𝐵 ∈ (1(,)+∞) → 𝐵 ∈ (ℂ ∖ {0, 1}))
50		recn 10139	. . . . . 6 ⊢ (𝑋 ∈ ℝ → 𝑋 ∈ ℂ)
51	50	adantr 472	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℂ)
52	17	gt0ne0d 10705	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ≠ 0)
53	51, 52	jca 555	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
54		eldifsn 4425	. . . 4 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
55	53, 3, 54	3imtr4i 281	. . 3 ⊢ (𝑋 ∈ (1[,)+∞) → 𝑋 ∈ (ℂ ∖ {0}))
56		logbval 24624	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝑋) = ((log‘𝑋) / (log‘𝐵)))
57	49, 55, 56	syl2an 495	. 2 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (𝐵 log_b 𝑋) = ((log‘𝑋) / (log‘𝐵)))
58	41, 57	breqtrrd 4788	1 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 ≤ (𝐵 log_b 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1596 ∈ wcel 2103 ≠ wne 2896 ∖ cdif 3677 {csn 4285 {cpr 4287 class class class wbr 4760 ‘cfv 6001 (class class class)co 6765 ℂcc 10047 ℝcr 10048 0cc0 10049 1c1 10050 +∞cpnf 10184 ℝ^*cxr 10186 < clt 10187 ≤ cle 10188 / cdiv 10797 ℝ⁺crp 11946 (,)cioo 12289 [,)cico 12291 logclog 24421 log_b clogb 24622

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-inf2 8651 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 ax-pre-sup 10127 ax-addf 10128 ax-mulf 10129

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-fal 1602 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-int 4584 df-iun 4630 df-iin 4631 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-se 5178 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-isom 6010 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-of 7014 df-om 7183 df-1st 7285 df-2nd 7286 df-supp 7416 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-1o 7680 df-2o 7681 df-oadd 7684 df-er 7862 df-map 7976 df-pm 7977 df-ixp 8026 df-en 8073 df-dom 8074 df-sdom 8075 df-fin 8076 df-fsupp 8392 df-fi 8433 df-sup 8464 df-inf 8465 df-oi 8531 df-card 8878 df-cda 9103 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-div 10798 df-nn 11134 df-2 11192 df-3 11193 df-4 11194 df-5 11195 df-6 11196 df-7 11197 df-8 11198 df-9 11199 df-n0 11406 df-z 11491 df-dec 11607 df-uz 11801 df-q 11903 df-rp 11947 df-xneg 12060 df-xadd 12061 df-xmul 12062 df-ioo 12293 df-ioc 12294 df-ico 12295 df-icc 12296 df-fz 12441 df-fzo 12581 df-fl 12708 df-mod 12784 df-seq 12917 df-exp 12976 df-fac 13176 df-bc 13205 df-hash 13233 df-shft 13927 df-cj 13959 df-re 13960 df-im 13961 df-sqrt 14095 df-abs 14096 df-limsup 14322 df-clim 14339 df-rlim 14340 df-sum 14537 df-ef 14918 df-sin 14920 df-cos 14921 df-pi 14923 df-struct 15982 df-ndx 15983 df-slot 15984 df-base 15986 df-sets 15987 df-ress 15988 df-plusg 16077 df-mulr 16078 df-starv 16079 df-sca 16080 df-vsca 16081 df-ip 16082 df-tset 16083 df-ple 16084 df-ds 16087 df-unif 16088 df-hom 16089 df-cco 16090 df-rest 16206 df-topn 16207 df-0g 16225 df-gsum 16226 df-topgen 16227 df-pt 16228 df-prds 16231 df-xrs 16285 df-qtop 16290 df-imas 16291 df-xps 16293 df-mre 16369 df-mrc 16370 df-acs 16372 df-mgm 17364 df-sgrp 17406 df-mnd 17417 df-submnd 17458 df-mulg 17663 df-cntz 17871 df-cmn 18316 df-psmet 19861 df-xmet 19862 df-met 19863 df-bl 19864 df-mopn 19865 df-fbas 19866 df-fg 19867 df-cnfld 19870 df-top 20822 df-topon 20839 df-topsp 20860 df-bases 20873 df-cld 20946 df-ntr 20947 df-cls 20948 df-nei 21025 df-lp 21063 df-perf 21064 df-cn 21154 df-cnp 21155 df-haus 21242 df-tx 21488 df-hmeo 21681 df-fil 21772 df-fm 21864 df-flim 21865 df-flf 21866 df-xms 22247 df-ms 22248 df-tms 22249 df-cncf 22803 df-limc 23750 df-dv 23751 df-log 24423 df-logb 24623

This theorem is referenced by: rege1logbzge0 42780

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