relogbmul - Metamath Proof Explorer

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Theorem relogbmul 27114

Description: The logarithm of the product of two positive real numbers is the sum of logarithms. Property 2 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 29-May-2020.)

**Assertion**
Ref	Expression
relogbmul	⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 · 𝐶)) = ((𝐵 log_b 𝐴) + (𝐵 log_b 𝐶)))

**Proof of Theorem relogbmul**
Step	Hyp	Ref	Expression
1		relogmul 26929	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘(𝐴 · 𝐶)) = ((log‘𝐴) + (log‘𝐶)))
2	1	adantl 487	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘(𝐴 · 𝐶)) = ((log‘𝐴) + (log‘𝐶)))
3	2	oveq1d 7439	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((log‘(𝐴 · 𝐶)) / (log‘𝐵)) = (((log‘𝐴) + (log‘𝐶)) / (log‘𝐵)))
4		relogcl 26912	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	recnd 11344	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
6	5	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐴) ∈ ℂ)
7		relogcl 26912	. . . . . 6 ⊢ (𝐶 ∈ ℝ⁺ → (log‘𝐶) ∈ ℝ)
8	7	recnd 11344	. . . . 5 ⊢ (𝐶 ∈ ℝ⁺ → (log‘𝐶) ∈ ℂ)
9	8	adantl 487	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐶) ∈ ℂ)
10		eldifpr 4615	. . . . . . . 8 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
11		3simpa 1166	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
12	10, 11	sylbi 220	. . . . . . 7 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
13		logcl 26905	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (log‘𝐵) ∈ ℂ)
14	12, 13	syl 18	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (log‘𝐵) ∈ ℂ)
15		logccne0 26915	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (log‘𝐵) ≠ 0)
16	10, 15	sylbi 220	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (log‘𝐵) ≠ 0)
17	14, 16	jca 521	. . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → ((log‘𝐵) ∈ ℂ ∧ (log‘𝐵) ≠ 0))
18	17	adantr 486	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((log‘𝐵) ∈ ℂ ∧ (log‘𝐵) ≠ 0))
19		divdir 12006	. . . 4 ⊢ (((log‘𝐴) ∈ ℂ ∧ (log‘𝐶) ∈ ℂ ∧ ((log‘𝐵) ∈ ℂ ∧ (log‘𝐵) ≠ 0)) → (((log‘𝐴) + (log‘𝐶)) / (log‘𝐵)) = (((log‘𝐴) / (log‘𝐵)) + ((log‘𝐶) / (log‘𝐵))))
20	6, 9, 18, 19	syl2an23an 1450	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (((log‘𝐴) + (log‘𝐶)) / (log‘𝐵)) = (((log‘𝐴) / (log‘𝐵)) + ((log‘𝐶) / (log‘𝐵))))
21	3, 20	eqtrd 2795	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((log‘(𝐴 · 𝐶)) / (log‘𝐵)) = (((log‘𝐴) / (log‘𝐵)) + ((log‘𝐶) / (log‘𝐵))))
22		rpcn 13138	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
23		rpcn 13138	. . . . 5 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ∈ ℂ)
24		mulcl 11291	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) ∈ ℂ)
25	22, 23, 24	syl2an 608	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · 𝐶) ∈ ℂ)
26	22	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
27	23	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
28		rpne0 13144	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
29	28	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ≠ 0)
30		rpne0 13144	. . . . . 6 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ≠ 0)
31	30	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ≠ 0)
32	26, 27, 29, 31	mulne0d 11975	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · 𝐶) ≠ 0)
33		eldifsn 4744	. . . 4 ⊢ ((𝐴 · 𝐶) ∈ (ℂ ∖ {0}) ↔ ((𝐴 · 𝐶) ∈ ℂ ∧ (𝐴 · 𝐶) ≠ 0))
34	25, 32, 33	sylanbrc 595	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · 𝐶) ∈ (ℂ ∖ {0}))
35		logbval 27103	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 · 𝐶) ∈ (ℂ ∖ {0})) → (𝐵 log_b (𝐴 · 𝐶)) = ((log‘(𝐴 · 𝐶)) / (log‘𝐵)))
36	34, 35	sylan2 605	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 · 𝐶)) = ((log‘(𝐴 · 𝐶)) / (log‘𝐵)))
37		rpcndif0 13148	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ (ℂ ∖ {0}))
38	37	adantr 486	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ (ℂ ∖ {0}))
39		logbval 27103	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝐴) = ((log‘𝐴) / (log‘𝐵)))
40	38, 39	sylan2 605	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) = ((log‘𝐴) / (log‘𝐵)))
41		rpcndif0 13148	. . . . 5 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ∈ (ℂ ∖ {0}))
42	41	adantl 487	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ (ℂ ∖ {0}))
43		logbval 27103	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐶 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝐶) = ((log‘𝐶) / (log‘𝐵)))
44	42, 43	sylan2 605	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) = ((log‘𝐶) / (log‘𝐵)))
45	40, 44	oveq12d 7442	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((𝐵 log_b 𝐴) + (𝐵 log_b 𝐶)) = (((log‘𝐴) / (log‘𝐵)) + ((log‘𝐶) / (log‘𝐵))))
46	21, 36, 45	3eqtr4d 2805	1 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 · 𝐶)) = ((𝐵 log_b 𝐴) + (𝐵 log_b 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∖ cdif 4044 {csn 4584 {cpr 4586 ‘cfv 6543 (class class class)co 7424 ℂcc 11205 0cc0 11207 1c1 11208 + caddc 11210 · cmul 11212 / cdiv 11980 ℝ⁺crp 13127 logclog 26891 log_b clogb 27101

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ioc 13488 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-fac 14425 df-bc 14454 df-hash 14482 df-shft 15229 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-limsup 15647 df-clim 15664 df-rlim 15665 df-sum 15863 df-ef 16242 df-sin 16244 df-cos 16245 df-pi 16247 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-fbas 21684 df-fg 21685 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cld 23346 df-ntr 23347 df-cls 23348 df-nei 23425 df-lp 23463 df-perf 23464 df-cn 23554 df-cnp 23555 df-haus 23642 df-tx 23890 df-hmeo 24083 df-fil 24174 df-fm 24266 df-flim 24267 df-flf 24268 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-limc 26195 df-dv 26196 df-log 26893 df-logb 27102

This theorem is used by: relogbmulexp 27115 blennnt2 49837

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