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Theorem rprelogbmul 15685

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 15599	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (log‘(𝐴 · 𝐶)) = ((log‘𝐴) + (log‘𝐶)))
2	1	adantl 277	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘(𝐴 · 𝐶)) = ((log‘𝐴) + (log‘𝐶)))
3	2	oveq1d 6033	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((log‘(𝐴 · 𝐶)) / (log‘𝐵)) = (((log‘𝐴) + (log‘𝐶)) / (log‘𝐵)))
4		relogcl 15592	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
5	4	recnd 8208	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
6	5	ad2antrl 490	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘𝐴) ∈ ℂ)
7		relogcl 15592	. . . . . 6 ⊢ (𝐶 ∈ ℝ⁺ → (log‘𝐶) ∈ ℝ)
8	7	recnd 8208	. . . . 5 ⊢ (𝐶 ∈ ℝ⁺ → (log‘𝐶) ∈ ℂ)
9	8	ad2antll 491	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘𝐶) ∈ ℂ)
10		simpll 527	. . . . . 6 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐵 ∈ ℝ⁺)
11	10	relogcld 15612	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘𝐵) ∈ ℝ)
12	11	recnd 8208	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘𝐵) ∈ ℂ)
13		simplr 529	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐵 # 1)
14	10, 13	logrpap0d 15608	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (log‘𝐵) # 0)
15	6, 9, 12, 14	divdirapd 9009	. . 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 2264	. 2 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((log‘(𝐴 · 𝐶)) / (log‘𝐵)) = (((log‘𝐴) / (log‘𝐵)) + ((log‘𝐶) / (log‘𝐵))))
17		rpmulcl 9913	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · 𝐶) ∈ ℝ⁺)
18	17	adantl 277	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐴 · 𝐶) ∈ ℝ⁺)
19		rplogbval 15675	. . 3 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ (𝐴 · 𝐶) ∈ ℝ⁺) → (𝐵 log_b (𝐴 · 𝐶)) = ((log‘(𝐴 · 𝐶)) / (log‘𝐵)))
20	10, 13, 18, 19	syl3anc 1273	. 2 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 · 𝐶)) = ((log‘(𝐴 · 𝐶)) / (log‘𝐵)))
21		simprl 531	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐴 ∈ ℝ⁺)
22		rplogbval 15675	. . . 4 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ 𝐴 ∈ ℝ⁺) → (𝐵 log_b 𝐴) = ((log‘𝐴) / (log‘𝐵)))
23	10, 13, 21, 22	syl3anc 1273	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) = ((log‘𝐴) / (log‘𝐵)))
24		simprr 533	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐶 ∈ ℝ⁺)
25		rplogbval 15675	. . . 4 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ 𝐶 ∈ ℝ⁺) → (𝐵 log_b 𝐶) = ((log‘𝐶) / (log‘𝐵)))
26	10, 13, 24, 25	syl3anc 1273	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) = ((log‘𝐶) / (log‘𝐵)))
27	23, 26	oveq12d 6036	. 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 2274	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 104 = wceq 1397 ∈ wcel 2202 class class class wbr 4088 ‘cfv 5326 (class class class)co 6018 ℂcc 8030 1c1 8033 + caddc 8035 · cmul 8037 # cap 8761 / cdiv 8852 ℝ⁺crp 9888 logclog 15586 log_b clogb 15673

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-pre-suploc 8153 ax-addf 8154 ax-mulf 8155

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-oadd 6586 df-er 6702 df-map 6819 df-pm 6820 df-en 6910 df-dom 6911 df-fin 6912 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-ioo 10127 df-ico 10129 df-icc 10130 df-fz 10244 df-fzo 10378 df-seqfrec 10711 df-exp 10802 df-fac 10989 df-bc 11011 df-ihash 11039 df-shft 11380 df-cj 11407 df-re 11408 df-im 11409 df-rsqrt 11563 df-abs 11564 df-clim 11844 df-sumdc 11919 df-ef 12214 df-e 12215 df-rest 13329 df-topgen 13348 df-psmet 14563 df-xmet 14564 df-met 14565 df-bl 14566 df-mopn 14567 df-top 14728 df-topon 14741 df-bases 14773 df-ntr 14826 df-cn 14918 df-cnp 14919 df-tx 14983 df-cncf 15301 df-limced 15386 df-dvap 15387 df-relog 15588 df-logb 15674

This theorem is referenced by: rprelogbmulexp 15686

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