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Theorem rprelogbmul 13232

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
rprelogbmul	⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 · 𝐶)) = ((𝐵 log_b 𝐴) + (𝐵 log_b 𝐶)))

**Proof of Theorem rprelogbmul**
Step	Hyp	Ref	Expression
1		relogmul 13150	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘(𝐴 · 𝐶)) = ((log‘𝐴) + (log‘𝐶)))
2	1	adantl 275	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘(𝐴 · 𝐶)) = ((log‘𝐴) + (log‘𝐶)))
3	2	oveq1d 5833	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((log‘(𝐴 · 𝐶)) / (log‘𝐵)) = (((log‘𝐴) + (log‘𝐶)) / (log‘𝐵)))
4		relogcl 13143	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	recnd 7889	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
6	5	ad2antrl 482	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘𝐴) ∈ ℂ)
7		relogcl 13143	. . . . . 6 ⊢ (𝐶 ∈ ℝ⁺ → (log‘𝐶) ∈ ℝ)
8	7	recnd 7889	. . . . 5 ⊢ (𝐶 ∈ ℝ⁺ → (log‘𝐶) ∈ ℂ)
9	8	ad2antll 483	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘𝐶) ∈ ℂ)
10		simpll 519	. . . . . 6 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐵 ∈ ℝ⁺)
11	10	relogcld 13163	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘𝐵) ∈ ℝ)
12	11	recnd 7889	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘𝐵) ∈ ℂ)
13		simplr 520	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐵 # 1)
14	10, 13	logrpap0d 13159	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘𝐵) # 0)
15	6, 9, 12, 14	divdirapd 8685	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (((log‘𝐴) + (log‘𝐶)) / (log‘𝐵)) = (((log‘𝐴) / (log‘𝐵)) + ((log‘𝐶) / (log‘𝐵))))
16	3, 15	eqtrd 2190	. 2 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((log‘(𝐴 · 𝐶)) / (log‘𝐵)) = (((log‘𝐴) / (log‘𝐵)) + ((log‘𝐶) / (log‘𝐵))))
17		rpmulcl 9567	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · 𝐶) ∈ ℝ⁺)
18	17	adantl 275	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐴 · 𝐶) ∈ ℝ⁺)
19		rplogbval 13222	. . 3 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ (𝐴 · 𝐶) ∈ ℝ⁺) → (𝐵 log_b (𝐴 · 𝐶)) = ((log‘(𝐴 · 𝐶)) / (log‘𝐵)))
20	10, 13, 18, 19	syl3anc 1220	. 2 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 · 𝐶)) = ((log‘(𝐴 · 𝐶)) / (log‘𝐵)))
21		simprl 521	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐴 ∈ ℝ⁺)
22		rplogbval 13222	. . . 4 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ 𝐴 ∈ ℝ⁺) → (𝐵 log_b 𝐴) = ((log‘𝐴) / (log‘𝐵)))
23	10, 13, 21, 22	syl3anc 1220	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) = ((log‘𝐴) / (log‘𝐵)))
24		simprr 522	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐶 ∈ ℝ⁺)
25		rplogbval 13222	. . . 4 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ 𝐶 ∈ ℝ⁺) → (𝐵 log_b 𝐶) = ((log‘𝐶) / (log‘𝐵)))
26	10, 13, 24, 25	syl3anc 1220	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) = ((log‘𝐶) / (log‘𝐵)))
27	23, 26	oveq12d 5836	. 2 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((𝐵 log_b 𝐴) + (𝐵 log_b 𝐶)) = (((log‘𝐴) / (log‘𝐵)) + ((log‘𝐶) / (log‘𝐵))))
28	16, 20, 27	3eqtr4d 2200	1 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 · 𝐶)) = ((𝐵 log_b 𝐴) + (𝐵 log_b 𝐶)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 ∈ wcel 2128 class class class wbr 3965 ‘cfv 5167 (class class class)co 5818 ℂcc 7713 1c1 7716 + caddc 7718 · cmul 7720 # cap 8439 / cdiv 8528 ℝ⁺crp 9542 logclog 13137 log_b clogb 13220

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4134 ax-pr 4168 ax-un 4392 ax-setind 4494 ax-iinf 4545 ax-cnex 7806 ax-resscn 7807 ax-1cn 7808 ax-1re 7809 ax-icn 7810 ax-addcl 7811 ax-addrcl 7812 ax-mulcl 7813 ax-mulrcl 7814 ax-addcom 7815 ax-mulcom 7816 ax-addass 7817 ax-mulass 7818 ax-distr 7819 ax-i2m1 7820 ax-0lt1 7821 ax-1rid 7822 ax-0id 7823 ax-rnegex 7824 ax-precex 7825 ax-cnre 7826 ax-pre-ltirr 7827 ax-pre-ltwlin 7828 ax-pre-lttrn 7829 ax-pre-apti 7830 ax-pre-ltadd 7831 ax-pre-mulgt0 7832 ax-pre-mulext 7833 ax-arch 7834 ax-caucvg 7835 ax-pre-suploc 7836 ax-addf 7837 ax-mulf 7838

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-disj 3943 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4252 df-po 4255 df-iso 4256 df-iord 4325 df-on 4327 df-ilim 4328 df-suc 4330 df-iom 4548 df-xp 4589 df-rel 4590 df-cnv 4591 df-co 4592 df-dm 4593 df-rn 4594 df-res 4595 df-ima 4596 df-iota 5132 df-fun 5169 df-fn 5170 df-f 5171 df-f1 5172 df-fo 5173 df-f1o 5174 df-fv 5175 df-isom 5176 df-riota 5774 df-ov 5821 df-oprab 5822 df-mpo 5823 df-of 6026 df-1st 6082 df-2nd 6083 df-recs 6246 df-irdg 6311 df-frec 6332 df-1o 6357 df-oadd 6361 df-er 6473 df-map 6588 df-pm 6589 df-en 6679 df-dom 6680 df-fin 6681 df-sup 6920 df-inf 6921 df-pnf 7897 df-mnf 7898 df-xr 7899 df-ltxr 7900 df-le 7901 df-sub 8031 df-neg 8032 df-reap 8433 df-ap 8440 df-div 8529 df-inn 8817 df-2 8875 df-3 8876 df-4 8877 df-n0 9074 df-z 9151 df-uz 9423 df-q 9511 df-rp 9543 df-xneg 9661 df-xadd 9662 df-ioo 9778 df-ico 9780 df-icc 9781 df-fz 9895 df-fzo 10024 df-seqfrec 10327 df-exp 10401 df-fac 10582 df-bc 10604 df-ihash 10632 df-shft 10697 df-cj 10724 df-re 10725 df-im 10726 df-rsqrt 10880 df-abs 10881 df-clim 11158 df-sumdc 11233 df-ef 11527 df-e 11528 df-rest 12313 df-topgen 12332 df-psmet 12347 df-xmet 12348 df-met 12349 df-bl 12350 df-mopn 12351 df-top 12356 df-topon 12369 df-bases 12401 df-ntr 12456 df-cn 12548 df-cnp 12549 df-tx 12613 df-cncf 12918 df-limced 12985 df-dvap 12986 df-relog 13139 df-logb 13221

This theorem is referenced by: rprelogbmulexp 13233

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