rege1logbrege0 - Mathbox for Alexander van der Vekens

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Theorem rege1logbrege0 48408

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 11259	. . . . . . 7 ⊢ 1 ∈ ℝ
2		elicopnf 13482	. . . . . . 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 26662	. . . 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 11783	. . . . . . . . . 10 ⊢ 0 < 1
11		0red 11262	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 0 ∈ ℝ)
12		1red 11260	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 1 ∈ ℝ)
13		id 22	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ℝ → 𝑋 ∈ ℝ)
14		ltletr 11351	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑋 ∈ ℝ) → ((0 < 1 ∧ 1 ≤ 𝑋) → 0 < 𝑋))
15	11, 12, 13, 14	syl3anc 1370	. . . . . . . . . 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 13072	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℝ⁺)
19	3, 18	sylbi 217	. . . . . 6 ⊢ (𝑋 ∈ (1[,)+∞) → 𝑋 ∈ ℝ⁺)
20	19	relogcld 26680	. . . . 5 ⊢ (𝑋 ∈ (1[,)+∞) → (log‘𝑋) ∈ ℝ)
21	20	adantl 481	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (log‘𝑋) ∈ ℝ)
22		1xr 11318	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
23		elioopnf 13480	. . . . . . . 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 11262	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 0 ∈ ℝ)
27		1red 11260	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 1 ∈ ℝ)
28		id 22	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ)
29		lttr 11335	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 < 1 ∧ 1 < 𝐵) → 0 < 𝐵))
30	26, 27, 28, 29	syl3anc 1370	. . . . . . . . . 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 13072	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℝ⁺)
34	24, 33	sylbi 217	. . . . . 6 ⊢ (𝐵 ∈ (1(,)+∞) → 𝐵 ∈ ℝ⁺)
35	34	relogcld 26680	. . . . 5 ⊢ (𝐵 ∈ (1(,)+∞) → (log‘𝐵) ∈ ℝ)
36	35	adantr 480	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → (log‘𝐵) ∈ ℝ)
37		regt1loggt0 48386	. . . . 5 ⊢ (𝐵 ∈ (1(,)+∞) → 0 < (log‘𝐵))
38	37	adantr 480	. . . 4 ⊢ ((𝐵 ∈ (1(,)+∞) ∧ 𝑋 ∈ (1[,)+∞)) → 0 < (log‘𝐵))
39		ge0div 12133	. . . 4 ⊢ (((log‘𝑋) ∈ ℝ ∧ (log‘𝐵) ∈ ℝ ∧ 0 < (log‘𝐵)) → (0 ≤ (log‘𝑋) ↔ 0 ≤ ((log‘𝑋) / (log‘𝐵))))
40	21, 36, 38, 39	syl3anc 1370	. . 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 11243	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
43	42	adantr 480	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ∈ ℂ)
44	32	gt0ne0d 11825	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → 𝐵 ≠ 0)
45	27, 28	ltlend 11404	. . . . . 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 1127	. . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 1 < 𝐵) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
48		eldifpr 4663	. . . 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 11243	. . . . . 6 ⊢ (𝑋 ∈ ℝ → 𝑋 ∈ ℂ)
51	50	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ∈ ℂ)
52	17	gt0ne0d 11825	. . . . 5 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → 𝑋 ≠ 0)
53	51, 52	jca 511	. . . 4 ⊢ ((𝑋 ∈ ℝ ∧ 1 ≤ 𝑋) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
54		eldifsn 4791	. . . 4 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
55	53, 3, 54	3imtr4i 292	. . 3 ⊢ (𝑋 ∈ (1[,)+∞) → 𝑋 ∈ (ℂ ∖ {0}))
56		logbval 26824	. . 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 5176	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 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 {cpr 4633 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 +∞cpnf 11290 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 / cdiv 11918 ℝ⁺crp 13032 (,)cioo 13384 [,)cico 13386 logclog 26611 log_b clogb 26822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-logb 26823

This theorem is referenced by: rege1logbzge0 48409

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