cxplogb - Metamath Proof Explorer

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Theorem cxplogb 26136

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 26116	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝑋) = ((log‘𝑋) / (log‘𝐵)))
2	1	oveq2d 7373	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑_𝑐(𝐵 log_b 𝑋)) = (𝐵↑_𝑐((log‘𝑋) / (log‘𝐵))))
3		eldifi 4086	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → 𝐵 ∈ ℂ)
4	3	adantr 481	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → 𝐵 ∈ ℂ)
5		eldif 3920	. . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ {0, 1}))
6		c0ex 11149	. . . . . . . . 9 ⊢ 0 ∈ V
7	6	prid1 4723	. . . . . . . 8 ⊢ 0 ∈ {0, 1}
8		eleq1 2825	. . . . . . . 8 ⊢ (𝐵 = 0 → (𝐵 ∈ {0, 1} ↔ 0 ∈ {0, 1}))
9	7, 8	mpbiri 257	. . . . . . 7 ⊢ (𝐵 = 0 → 𝐵 ∈ {0, 1})
10	9	necon3bi 2970	. . . . . 6 ⊢ (¬ 𝐵 ∈ {0, 1} → 𝐵 ≠ 0)
11	10	adantl 482	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ {0, 1}) → 𝐵 ≠ 0)
12	5, 11	sylbi 216	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → 𝐵 ≠ 0)
13	12	adantr 481	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → 𝐵 ≠ 0)
14		eldif 3920	. . . . . . 7 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ ¬ 𝑋 ∈ {0}))
15	6	snid 4622	. . . . . . . . . 10 ⊢ 0 ∈ {0}
16		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑋 = 0 → (𝑋 ∈ {0} ↔ 0 ∈ {0}))
17	15, 16	mpbiri 257	. . . . . . . . 9 ⊢ (𝑋 = 0 → 𝑋 ∈ {0})
18	17	necon3bi 2970	. . . . . . . 8 ⊢ (¬ 𝑋 ∈ {0} → 𝑋 ≠ 0)
19	18	anim2i 617	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 ∈ {0}) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
20	14, 19	sylbi 216	. . . . . 6 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
21		logcl 25924	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (log‘𝑋) ∈ ℂ)
22	20, 21	syl 17	. . . . 5 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (log‘𝑋) ∈ ℂ)
23	22	adantl 482	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝑋) ∈ ℂ)
24	10	anim2i 617	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ {0, 1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
25	5, 24	sylbi 216	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
26		logcl 25924	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (log‘𝐵) ∈ ℂ)
27	25, 26	syl 17	. . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (log‘𝐵) ∈ ℂ)
28	27	adantr 481	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝐵) ∈ ℂ)
29		eldifpr 4618	. . . . . . 7 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
30	29	biimpi 215	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
31	30	adantr 481	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
32		logccne0 25934	. . . . 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 11931	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ((log‘𝑋) / (log‘𝐵)) ∈ ℂ)
35	4, 13, 34	cxpefd 26067	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑_𝑐((log‘𝑋) / (log‘𝐵))) = (exp‘(((log‘𝑋) / (log‘𝐵)) · (log‘𝐵))))
36		eldifsn 4747	. . . . . . 7 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
37	36, 21	sylbi 216	. . . . . 6 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (log‘𝑋) ∈ ℂ)
38	37	adantl 482	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝑋) ∈ ℂ)
39	29, 32	sylbi 216	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (log‘𝐵) ≠ 0)
40	39	adantr 481	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝐵) ≠ 0)
41	38, 28, 40	divcan1d 11932	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (((log‘𝑋) / (log‘𝐵)) · (log‘𝐵)) = (log‘𝑋))
42	41	fveq2d 6846	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (exp‘(((log‘𝑋) / (log‘𝐵)) · (log‘𝐵))) = (exp‘(log‘𝑋)))
43		eflog 25932	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (exp‘(log‘𝑋)) = 𝑋)
44	36, 43	sylbi 216	. . . 4 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (exp‘(log‘𝑋)) = 𝑋)
45	44	adantl 482	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (exp‘(log‘𝑋)) = 𝑋)
46	42, 45	eqtrd 2776	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (exp‘(((log‘𝑋) / (log‘𝐵)) · (log‘𝐵))) = 𝑋)
47	2, 35, 46	3eqtrd 2780	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3907 {csn 4586 {cpr 4588 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 0cc0 11051 1c1 11052 · cmul 11056 / cdiv 11812 expce 15944 logclog 25910 ↑_𝑐ccxp 25911 log_b clogb 26114

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-cxp 25913 df-logb 26115

This theorem is referenced by: relogbcxpb 26137 logbgcd1irr 26144 sqrt2cxp2logb9e3 26149 aks4d1p1p4 40528 aks4d1p6 40538 fllogbd 46636 nnpw2blen 46656 dignn0ldlem 46678

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