relogbcl - Metamath Proof Explorer

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Theorem relogbcl 25463

Description: Closure of the general logarithm with a positive real base on positive reals. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)

**Assertion**
Ref	Expression
relogbcl	⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → (𝐵 log_b 𝑋) ∈ ℝ)

**Proof of Theorem relogbcl**
Step	Hyp	Ref	Expression
1		simp1 1133	. . . . . 6 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → 𝐵 ∈ ℝ⁺)
2	1	rpcnne0d 12486	. . . . 5 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
3		simp3 1135	. . . . 5 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → 𝐵 ≠ 1)
4		df-3an 1086	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) ↔ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ 𝐵 ≠ 1))
5	2, 3, 4	sylanbrc 586	. . . 4 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
6		eldifpr 4557	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
7	5, 6	sylibr 237	. . 3 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → 𝐵 ∈ (ℂ ∖ {0, 1}))
8		simp2 1134	. . . . 5 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → 𝑋 ∈ ℝ⁺)
9	8	rpcnne0d 12486	. . . 4 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
10		eldifsn 4680	. . . 4 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
11	9, 10	sylibr 237	. . 3 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → 𝑋 ∈ (ℂ ∖ {0}))
12		logbval 25456	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝑋) = ((log‘𝑋) / (log‘𝐵)))
13	7, 11, 12	syl2anc 587	. 2 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → (𝐵 log_b 𝑋) = ((log‘𝑋) / (log‘𝐵)))
14		relogcl 25271	. . . 4 ⊢ (𝑋 ∈ ℝ⁺ → (log‘𝑋) ∈ ℝ)
15	14	3ad2ant2 1131	. . 3 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → (log‘𝑋) ∈ ℝ)
16		relogcl 25271	. . . 4 ⊢ (𝐵 ∈ ℝ⁺ → (log‘𝐵) ∈ ℝ)
17	16	3ad2ant1 1130	. . 3 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → (log‘𝐵) ∈ ℝ)
18		logne0 25275	. . . 4 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → (log‘𝐵) ≠ 0)
19	18	3adant2 1128	. . 3 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → (log‘𝐵) ≠ 0)
20	15, 17, 19	redivcld 11511	. 2 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → ((log‘𝑋) / (log‘𝐵)) ∈ ℝ)
21	13, 20	eqeltrd 2852	1 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝑋 ∈ ℝ⁺ ∧ 𝐵 ≠ 1) → (𝐵 log_b 𝑋) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2951 ∖ cdif 3857 {csn 4525 {cpr 4527 ‘cfv 6339 (class class class)co 7155 ℂcc 10578 ℝcr 10579 0cc0 10580 1c1 10581 / cdiv 11340 ℝ⁺crp 12435 logclog 25250 log_b clogb 25454

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-inf2 9142 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 ax-pre-sup 10658 ax-addf 10659 ax-mulf 10660

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-se 5487 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7410 df-om 7585 df-1st 7698 df-2nd 7699 df-supp 7841 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-1o 8117 df-2o 8118 df-er 8304 df-map 8423 df-pm 8424 df-ixp 8485 df-en 8533 df-dom 8534 df-sdom 8535 df-fin 8536 df-fsupp 8872 df-fi 8913 df-sup 8944 df-inf 8945 df-oi 9012 df-card 9406 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-div 11341 df-nn 11680 df-2 11742 df-3 11743 df-4 11744 df-5 11745 df-6 11746 df-7 11747 df-8 11748 df-9 11749 df-n0 11940 df-z 12026 df-dec 12143 df-uz 12288 df-q 12394 df-rp 12436 df-xneg 12553 df-xadd 12554 df-xmul 12555 df-ioo 12788 df-ioc 12789 df-ico 12790 df-icc 12791 df-fz 12945 df-fzo 13088 df-fl 13216 df-mod 13292 df-seq 13424 df-exp 13485 df-fac 13689 df-bc 13718 df-hash 13746 df-shft 14479 df-cj 14511 df-re 14512 df-im 14513 df-sqrt 14647 df-abs 14648 df-limsup 14881 df-clim 14898 df-rlim 14899 df-sum 15096 df-ef 15474 df-sin 15476 df-cos 15477 df-pi 15479 df-struct 16548 df-ndx 16549 df-slot 16550 df-base 16552 df-sets 16553 df-ress 16554 df-plusg 16641 df-mulr 16642 df-starv 16643 df-sca 16644 df-vsca 16645 df-ip 16646 df-tset 16647 df-ple 16648 df-ds 16650 df-unif 16651 df-hom 16652 df-cco 16653 df-rest 16759 df-topn 16760 df-0g 16778 df-gsum 16779 df-topgen 16780 df-pt 16781 df-prds 16784 df-xrs 16838 df-qtop 16843 df-imas 16844 df-xps 16846 df-mre 16920 df-mrc 16921 df-acs 16923 df-mgm 17923 df-sgrp 17972 df-mnd 17983 df-submnd 18028 df-mulg 18297 df-cntz 18519 df-cmn 18980 df-psmet 20163 df-xmet 20164 df-met 20165 df-bl 20166 df-mopn 20167 df-fbas 20168 df-fg 20169 df-cnfld 20172 df-top 21599 df-topon 21616 df-topsp 21638 df-bases 21651 df-cld 21724 df-ntr 21725 df-cls 21726 df-nei 21803 df-lp 21841 df-perf 21842 df-cn 21932 df-cnp 21933 df-haus 22020 df-tx 22267 df-hmeo 22460 df-fil 22551 df-fm 22643 df-flim 22644 df-flf 22645 df-xms 23027 df-ms 23028 df-tms 23029 df-cncf 23584 df-limc 24570 df-dv 24571 df-log 25252 df-logb 25455

This theorem is referenced by: relogbzcl 25464 relogbf 25481 logbgcd1irr 25484 relogbcld 39565 logbpw2m1 45374 fllog2 45375 blennnelnn 45383 nnpw2blen 45387 blen1b 45395 blennnt2 45396 nnolog2flm1 45397 blennngt2o2 45399 blennn0e2 45401

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