relogbmul - Metamath Proof Explorer

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Theorem relogbmul 26835

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 26649	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘(𝐴 · 𝐶)) = ((log‘𝐴) + (log‘𝐶)))
2	1	adantl 481	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘(𝐴 · 𝐶)) = ((log‘𝐴) + (log‘𝐶)))
3	2	oveq1d 7446	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((log‘(𝐴 · 𝐶)) / (log‘𝐵)) = (((log‘𝐴) + (log‘𝐶)) / (log‘𝐵)))
4		relogcl 26632	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	recnd 11287	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
6	5	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐴) ∈ ℂ)
7		relogcl 26632	. . . . . 6 ⊢ (𝐶 ∈ ℝ⁺ → (log‘𝐶) ∈ ℝ)
8	7	recnd 11287	. . . . 5 ⊢ (𝐶 ∈ ℝ⁺ → (log‘𝐶) ∈ ℂ)
9	8	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘𝐶) ∈ ℂ)
10		eldifpr 4663	. . . . . . . 8 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1))
11		3simpa 1147	. . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
12	10, 11	sylbi 217	. . . . . . 7 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
13		logcl 26625	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (log‘𝐵) ∈ ℂ)
14	12, 13	syl 17	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (log‘𝐵) ∈ ℂ)
15		logccne0 26635	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1) → (log‘𝐵) ≠ 0)
16	10, 15	sylbi 217	. . . . . 6 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → (log‘𝐵) ≠ 0)
17	14, 16	jca 511	. . . . 5 ⊢ (𝐵 ∈ (ℂ ∖ {0, 1}) → ((log‘𝐵) ∈ ℂ ∧ (log‘𝐵) ≠ 0))
18	17	adantr 480	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((log‘𝐵) ∈ ℂ ∧ (log‘𝐵) ≠ 0))
19		divdir 11945	. . . 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 1422	. . 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 2775	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((log‘(𝐴 · 𝐶)) / (log‘𝐵)) = (((log‘𝐴) / (log‘𝐵)) + ((log‘𝐶) / (log‘𝐵))))
22		rpcn 13043	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
23		rpcn 13043	. . . . 5 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ∈ ℂ)
24		mulcl 11237	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · 𝐶) ∈ ℂ)
25	22, 23, 24	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · 𝐶) ∈ ℂ)
26	22	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
27	23	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
28		rpne0 13049	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
29	28	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ≠ 0)
30		rpne0 13049	. . . . . 6 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ≠ 0)
31	30	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ≠ 0)
32	26, 27, 29, 31	mulne0d 11913	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · 𝐶) ≠ 0)
33		eldifsn 4791	. . . 4 ⊢ ((𝐴 · 𝐶) ∈ (ℂ ∖ {0}) ↔ ((𝐴 · 𝐶) ∈ ℂ ∧ (𝐴 · 𝐶) ≠ 0))
34	25, 32, 33	sylanbrc 583	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · 𝐶) ∈ (ℂ ∖ {0}))
35		logbval 26824	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 · 𝐶) ∈ (ℂ ∖ {0})) → (𝐵 log_b (𝐴 · 𝐶)) = ((log‘(𝐴 · 𝐶)) / (log‘𝐵)))
36	34, 35	sylan2 593	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 · 𝐶)) = ((log‘(𝐴 · 𝐶)) / (log‘𝐵)))
37		rpcndif0 13052	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ (ℂ ∖ {0}))
38	37	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ (ℂ ∖ {0}))
39		logbval 26824	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝐴) = ((log‘𝐴) / (log‘𝐵)))
40	38, 39	sylan2 593	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) = ((log‘𝐴) / (log‘𝐵)))
41		rpcndif0 13052	. . . . 5 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ∈ (ℂ ∖ {0}))
42	41	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ (ℂ ∖ {0}))
43		logbval 26824	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐶 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝐶) = ((log‘𝐶) / (log‘𝐵)))
44	42, 43	sylan2 593	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) = ((log‘𝐶) / (log‘𝐵)))
45	40, 44	oveq12d 7449	. 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 2785	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

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 {cpr 4633 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 / cdiv 11918 ℝ⁺crp 13032 logclog 26611 log_b clogb 26822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 df-logb 26823

This theorem is referenced by: relogbmulexp 26836 blennnt2 48439

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