cxplogb - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cxplogb		Structured version Visualization version GIF version

Theorem cxplogb 26828

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 26808	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝑋) = ((log‘𝑋) / (log‘𝐵)))
2	1	oveq2d 7408	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑_𝑐(𝐵 log_b 𝑋)) = (𝐵↑_𝑐((log‘𝑋) / (log‘𝐵))))
3		eldifi 4084	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → 𝐵 ∈ ℂ)
4	3	adantr 484	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → 𝐵 ∈ ℂ)
5		eldif 3914	. . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ {0, 1}))
6		c0ex 11170	. . . . . . . . 9 ⊢ 0 ∈ V
7	6	prid1 4720	. . . . . . . 8 ⊢ 0 ∈ {0, 1}
8		eleq1 2849	. . . . . . . 8 ⊢ (𝐵 = 0 → (𝐵 ∈ {0, 1} ↔ 0 ∈ {0, 1}))
9	7, 8	mpbiri 260	. . . . . . 7 ⊢ (𝐵 = 0 → 𝐵 ∈ {0, 1})
10	9	necon3bi 2982	. . . . . 6 ⊢ (¬ 𝐵 ∈ {0, 1} → 𝐵 ≠ 0)
11	10	adantl 485	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ {0, 1}) → 𝐵 ≠ 0)
12	5, 11	sylbi 219	. . . 4 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → 𝐵 ≠ 0)
13	12	adantr 484	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → 𝐵 ≠ 0)
14		eldif 3914	. . . . . . 7 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ ¬ 𝑋 ∈ {0}))
15	6	snid 4620	. . . . . . . . . 10 ⊢ 0 ∈ {0}
16		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑋 = 0 → (𝑋 ∈ {0} ↔ 0 ∈ {0}))
17	15, 16	mpbiri 260	. . . . . . . . 9 ⊢ (𝑋 = 0 → 𝑋 ∈ {0})
18	17	necon3bi 2982	. . . . . . . 8 ⊢ (¬ 𝑋 ∈ {0} → 𝑋 ≠ 0)
19	18	anim2i 626	. . . . . . 7 ⊢ ((𝑋 ∈ ℂ ∧ ¬ 𝑋 ∈ {0}) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
20	14, 19	sylbi 219	. . . . . 6 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
21		logcl 26610	. . . . . 6 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (log‘𝑋) ∈ ℂ)
22	20, 21	syl 17	. . . . 5 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (log‘𝑋) ∈ ℂ)
23	22	adantl 485	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝑋) ∈ ℂ)
24	10	anim2i 626	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ {0, 1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
25	5, 24	sylbi 219	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
26		logcl 26610	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (log‘𝐵) ∈ ℂ)
27	25, 26	syl 17	. . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (log‘𝐵) ∈ ℂ)
28	27	adantr 484	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝐵) ∈ ℂ)
29		eldifpr 4616	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
30	29	birani 507	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
31		logccne0 26620	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (log‘𝐵) ≠ 0)
32	30, 31	syl 17	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝐵) ≠ 0)
33	23, 28, 32	divcld 11964	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → ((log‘𝑋) / (log‘𝐵)) ∈ ℂ)
34	4, 13, 33	cxpefd 26754	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵↑_𝑐((log‘𝑋) / (log‘𝐵))) = (exp‘(((log‘𝑋) / (log‘𝐵)) · (log‘𝐵))))
35		eldifsn 4745	. . . . . . 7 ⊢ (𝑋 ∈ (ℂ ∖ {0}) ↔ (𝑋 ∈ ℂ ∧ 𝑋 ≠ 0))
36	35, 21	sylbi 219	. . . . . 6 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (log‘𝑋) ∈ ℂ)
37	36	adantl 485	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝑋) ∈ ℂ)
38	29, 31	sylbi 219	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (log‘𝐵) ≠ 0)
39	38	adantr 484	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (log‘𝐵) ≠ 0)
40	37, 28, 39	divcan1d 11965	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (((log‘𝑋) / (log‘𝐵)) · (log‘𝐵)) = (log‘𝑋))
41	40	fveq2d 6867	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (exp‘(((log‘𝑋) / (log‘𝐵)) · (log‘𝐵))) = (exp‘(log‘𝑋)))
42		eflog 26618	. . . . 5 ⊢ ((𝑋 ∈ ℂ ∧ 𝑋 ≠ 0) → (exp‘(log‘𝑋)) = 𝑋)
43	35, 42	sylbi 219	. . . 4 ⊢ (𝑋 ∈ (ℂ ∖ {0}) → (exp‘(log‘𝑋)) = 𝑋)
44	43	adantl 485	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (exp‘(log‘𝑋)) = 𝑋)
45	41, 44	eqtrd 2796	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (exp‘(((log‘𝑋) / (log‘𝐵)) · (log‘𝐵))) = 𝑋)
46	2, 34, 45	3eqtrd 2800	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 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∖ cdif 3901 {csn 4581 {cpr 4583 ‘cfv 6517 (class class class)co 7392 ℂcc 11068 0cc0 11070 1c1 11071 · cmul 11075 / cdiv 11841 expce 16074 logclog 26596 ↑_𝑐ccxp 26597 log_b clogb 26806

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148 ax-addf 11149

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-q 12947 df-rp 12991 df-xneg 13111 df-xadd 13112 df-xmul 13113 df-ioo 13350 df-ioc 13351 df-ico 13352 df-icc 13353 df-fz 13510 df-fzo 13657 df-fl 13799 df-mod 13877 df-seq 14012 df-exp 14072 df-fac 14284 df-bc 14313 df-hash 14341 df-shft 15077 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-limsup 15481 df-clim 15498 df-rlim 15499 df-sum 15697 df-ef 16080 df-sin 16082 df-cos 16083 df-pi 16085 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17515 df-qtop 17520 df-imas 17521 df-xps 17523 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-mulg 19093 df-cntz 19340 df-cmn 19805 df-psmet 21396 df-xmet 21397 df-met 21398 df-bl 21399 df-mopn 21400 df-fbas 21401 df-fg 21402 df-cnfld 21405 df-top 22934 df-topon 22951 df-topsp 22973 df-bases 22986 df-cld 23059 df-ntr 23060 df-cls 23061 df-nei 23138 df-lp 23176 df-perf 23177 df-cn 23267 df-cnp 23268 df-haus 23355 df-tx 23602 df-hmeo 23795 df-fil 23886 df-fm 23978 df-flim 23979 df-flf 23980 df-xms 24360 df-ms 24361 df-tms 24362 df-cncf 24920 df-limc 25908 df-dv 25909 df-log 26598 df-cxp 26599 df-logb 26807

This theorem is referenced by: relogbcxpb 26829 logbgcd1irr 26836 sqrt2cxp2logb9e3 26841 aks4d1p1p4 42652 aks4d1p6 42662 aks6d1c7lem1 42761 fllogbd 49146 nnpw2blen 49166 dignn0ldlem 49188

W3C validator