cxplogb - Metamath Proof Explorer

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Theorem cxplogb 24719

Description: Identity law for the general logarithm. (Contributed by AV, 22-May-2020.)

**Assertion**
Ref	Expression
cxplogb	⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑_𝑐(𝐵 log_b 𝑋)) = 𝑋)

**Proof of Theorem cxplogb**
Step	Hyp	Ref	Expression
1		logbval 24699	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝑋) = ((log‘𝑋) / (log‘𝐵)))
2	1	oveq2d 6825	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑_𝑐(𝐵 log_b 𝑋)) = (𝐵↑_𝑐((log‘𝑋) / (log‘𝐵))))
3		eldifi 3871	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → 𝐵 ∈ ℂ)
4	3	adantr 472	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → 𝐵 ∈ ℂ)
5		eldif 3721	. . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ {0, 1}))
6		c0ex 10222	. . . . . . . . 9 ⊢ 0 ∈ V
7	6	prid1 4437	. . . . . . . 8 ⊢ 0 ∈ {0, 1}
8		eleq1 2823	. . . . . . . 8 ⊢ (𝐵 = 0 → (𝐵 ∈ {0, 1} ↔ 0 ∈ {0, 1}))
9	7, 8	mpbiri 248	. . . . . . 7 ⊢ (𝐵 = 0 → 𝐵 ∈ {0, 1})
10	9	necon3bi 2954	. . . . . 6 ⊢ (¬ 𝐵 ∈ {0, 1} → 𝐵 ≠ 0)
11	10	adantl 473	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ {0, 1}) → 𝐵 ≠ 0)
12	5, 11	sylbi 207	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → 𝐵 ≠ 0)
13	12	adantr 472	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → 𝐵 ≠ 0)
14		eldif 3721	. . . . . . 7 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ ¬ 𝑋 ∈ {0}))
15	6	snid 4349	. . . . . . . . . 10 ⊢ 0 ∈ {0}
16		eleq1 2823	. . . . . . . . . 10 ⊢ (𝑋 = 0 → (𝑋 ∈ {0} ↔ 0 ∈ {0}))
17	15, 16	mpbiri 248	. . . . . . . . 9 ⊢ (𝑋 = 0 → 𝑋 ∈ {0})
18	17	necon3bi 2954	. . . . . . . 8 ⊢ (¬ 𝑋 ∈ {0} → 𝑋 ≠ 0)
19	18	anim2i 594	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 ∈ {0}) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
20	14, 19	sylbi 207	. . . . . 6 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
21		logcl 24510	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (log‘𝑋) ∈ ℂ)
22	20, 21	syl 17	. . . . 5 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (log‘𝑋) ∈ ℂ)
23	22	adantl 473	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝑋) ∈ ℂ)
24	10	anim2i 594	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ {0, 1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
25	5, 24	sylbi 207	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
26		logcl 24510	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (log‘𝐵) ∈ ℂ)
27	25, 26	syl 17	. . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (log‘𝐵) ∈ ℂ)
28	27	adantr 472	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝐵) ∈ ℂ)
29		eldifpr 4345	. . . . . . 7 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
30	29	biimpi 206	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
31	30	adantr 472	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
32		logccne0 24520	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (log‘𝐵) ≠ 0)
33	31, 32	syl 17	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝐵) ≠ 0)
34	23, 28, 33	divcld 10989	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ((log‘𝑋) / (log‘𝐵)) ∈ ℂ)
35	4, 13, 34	cxpefd 24653	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑_𝑐((log‘𝑋) / (log‘𝐵))) = (exp‘(((log‘𝑋) / (log‘𝐵)) · (log‘𝐵))))
36		eldifsn 4458	. . . . . . 7 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
37	36, 21	sylbi 207	. . . . . 6 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (log‘𝑋) ∈ ℂ)
38	37	adantl 473	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝑋) ∈ ℂ)
39	29, 32	sylbi 207	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (log‘𝐵) ≠ 0)
40	39	adantr 472	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝐵) ≠ 0)
41	38, 28, 40	divcan1d 10990	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (((log‘𝑋) / (log‘𝐵)) · (log‘𝐵)) = (log‘𝑋))
42	41	fveq2d 6352	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (exp‘(((log‘𝑋) / (log‘𝐵)) · (log‘𝐵))) = (exp‘(log‘𝑋)))
43		eflog 24518	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (exp‘(log‘𝑋)) = 𝑋)
44	36, 43	sylbi 207	. . . 4 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (exp‘(log‘𝑋)) = 𝑋)
45	44	adantl 473	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (exp‘(log‘𝑋)) = 𝑋)
46	42, 45	eqtrd 2790	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (exp‘(((log‘𝑋) / (log‘𝐵)) · (log‘𝐵))) = 𝑋)
47	2, 35, 46	3eqtrd 2794	1 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑_𝑐(𝐵 log_b 𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1628 ∈ wcel 2135 ≠ wne 2928 ∖ cdif 3708 {csn 4317 {cpr 4319 ‘cfv 6045 (class class class)co 6809 ℂcc 10122 0cc0 10124 1c1 10125 · cmul 10129 / cdiv 10872 expce 14987 logclog 24496 ↑_𝑐ccxp 24497 log_b clogb 24697

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1867 ax-4 1882 ax-5 1984 ax-6 2050 ax-7 2086 ax-8 2137 ax-9 2144 ax-10 2164 ax-11 2179 ax-12 2192 ax-13 2387 ax-ext 2736 ax-rep 4919 ax-sep 4929 ax-nul 4937 ax-pow 4988 ax-pr 5051 ax-un 7110 ax-inf2 8707 ax-cnex 10180 ax-resscn 10181 ax-1cn 10182 ax-icn 10183 ax-addcl 10184 ax-addrcl 10185 ax-mulcl 10186 ax-mulrcl 10187 ax-mulcom 10188 ax-addass 10189 ax-mulass 10190 ax-distr 10191 ax-i2m1 10192 ax-1ne0 10193 ax-1rid 10194 ax-rnegex 10195 ax-rrecex 10196 ax-cnre 10197 ax-pre-lttri 10198 ax-pre-lttrn 10199 ax-pre-ltadd 10200 ax-pre-mulgt0 10201 ax-pre-sup 10202 ax-addf 10203 ax-mulf 10204

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1631 df-fal 1634 df-ex 1850 df-nf 1855 df-sb 2043 df-eu 2607 df-mo 2608 df-clab 2743 df-cleq 2749 df-clel 2752 df-nfc 2887 df-ne 2929 df-nel 3032 df-ral 3051 df-rex 3052 df-reu 3053 df-rmo 3054 df-rab 3055 df-v 3338 df-sbc 3573 df-csb 3671 df-dif 3714 df-un 3716 df-in 3718 df-ss 3725 df-pss 3727 df-nul 4055 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4585 df-int 4624 df-iun 4670 df-iin 4671 df-br 4801 df-opab 4861 df-mpt 4878 df-tr 4901 df-id 5170 df-eprel 5175 df-po 5183 df-so 5184 df-fr 5221 df-se 5222 df-we 5223 df-xp 5268 df-rel 5269 df-cnv 5270 df-co 5271 df-dm 5272 df-rn 5273 df-res 5274 df-ima 5275 df-pred 5837 df-ord 5883 df-on 5884 df-lim 5885 df-suc 5886 df-iota 6008 df-fun 6047 df-fn 6048 df-f 6049 df-f1 6050 df-fo 6051 df-f1o 6052 df-fv 6053 df-isom 6054 df-riota 6770 df-ov 6812 df-oprab 6813 df-mpt2 6814 df-of 7058 df-om 7227 df-1st 7329 df-2nd 7330 df-supp 7460 df-wrecs 7572 df-recs 7633 df-rdg 7671 df-1o 7725 df-2o 7726 df-oadd 7729 df-er 7907 df-map 8021 df-pm 8022 df-ixp 8071 df-en 8118 df-dom 8119 df-sdom 8120 df-fin 8121 df-fsupp 8437 df-fi 8478 df-sup 8509 df-inf 8510 df-oi 8576 df-card 8951 df-cda 9178 df-pnf 10264 df-mnf 10265 df-xr 10266 df-ltxr 10267 df-le 10268 df-sub 10456 df-neg 10457 df-div 10873 df-nn 11209 df-2 11267 df-3 11268 df-4 11269 df-5 11270 df-6 11271 df-7 11272 df-8 11273 df-9 11274 df-n0 11481 df-z 11566 df-dec 11682 df-uz 11876 df-q 11978 df-rp 12022 df-xneg 12135 df-xadd 12136 df-xmul 12137 df-ioo 12368 df-ioc 12369 df-ico 12370 df-icc 12371 df-fz 12516 df-fzo 12656 df-fl 12783 df-mod 12859 df-seq 12992 df-exp 13051 df-fac 13251 df-bc 13280 df-hash 13308 df-shft 14002 df-cj 14034 df-re 14035 df-im 14036 df-sqrt 14170 df-abs 14171 df-limsup 14397 df-clim 14414 df-rlim 14415 df-sum 14612 df-ef 14993 df-sin 14995 df-cos 14996 df-pi 14998 df-struct 16057 df-ndx 16058 df-slot 16059 df-base 16061 df-sets 16062 df-ress 16063 df-plusg 16152 df-mulr 16153 df-starv 16154 df-sca 16155 df-vsca 16156 df-ip 16157 df-tset 16158 df-ple 16159 df-ds 16162 df-unif 16163 df-hom 16164 df-cco 16165 df-rest 16281 df-topn 16282 df-0g 16300 df-gsum 16301 df-topgen 16302 df-pt 16303 df-prds 16306 df-xrs 16360 df-qtop 16365 df-imas 16366 df-xps 16368 df-mre 16444 df-mrc 16445 df-acs 16447 df-mgm 17439 df-sgrp 17481 df-mnd 17492 df-submnd 17533 df-mulg 17738 df-cntz 17946 df-cmn 18391 df-psmet 19936 df-xmet 19937 df-met 19938 df-bl 19939 df-mopn 19940 df-fbas 19941 df-fg 19942 df-cnfld 19945 df-top 20897 df-topon 20914 df-topsp 20935 df-bases 20948 df-cld 21021 df-ntr 21022 df-cls 21023 df-nei 21100 df-lp 21138 df-perf 21139 df-cn 21229 df-cnp 21230 df-haus 21317 df-tx 21563 df-hmeo 21756 df-fil 21847 df-fm 21939 df-flim 21940 df-flf 21941 df-xms 22322 df-ms 22323 df-tms 22324 df-cncf 22878 df-limc 23825 df-dv 23826 df-log 24498 df-cxp 24499 df-logb 24698

This theorem is referenced by: relogbcxpb 24720 fllogbd 42860 nnpw2blen 42880 dignn0ldlem 42902

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