rprelogbdiv - Intuitionistic Logic Explorer

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Theorem rprelogbdiv 13515

Description: The logarithm of the quotient of two positive real numbers is the difference of logarithms. Property 3 of [Cohen4] p. 361. (Contributed by AV, 29-May-2020.)

**Assertion**
Ref	Expression
rprelogbdiv	⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 / 𝐶)) = ((𝐵 log_b 𝐴) − (𝐵 log_b 𝐶)))

**Proof of Theorem rprelogbdiv**
Step	Hyp	Ref	Expression
1		neg1rr 8963	. . 3 ⊢ -1 ∈ ℝ
2		rprelogbmulexp 13514	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺ ∧ -1 ∈ ℝ)) → (𝐵 log_b (𝐴 · (𝐶↑_𝑐-1))) = ((𝐵 log_b 𝐴) + (-1 · (𝐵 log_b 𝐶))))
3	1, 2	mp3anr3 1326	. 2 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 · (𝐶↑_𝑐-1))) = ((𝐵 log_b 𝐴) + (-1 · (𝐵 log_b 𝐶))))
4		rpcn 9598	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
5	4	adantr 274	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
6		rpcn 9598	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ∈ ℂ)
7	6	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
8		rpap0 9606	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 # 0)
9	8	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 # 0)
10	5, 7, 9	divrecapd 8689	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶)))
11		ax-1cn 7846	. . . . . . . . 9 ⊢ 1 ∈ ℂ
12		rpcxpneg 13468	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 1 ∈ ℂ) → (𝐶↑_𝑐-1) = (1 / (𝐶↑_𝑐1)))
13	11, 12	mpan2 422	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ⁺ → (𝐶↑_𝑐-1) = (1 / (𝐶↑_𝑐1)))
14		rpcxp1 13460	. . . . . . . . 9 ⊢ (𝐶 ∈ ℝ⁺ → (𝐶↑_𝑐1) = 𝐶)
15	14	oveq2d 5858	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ⁺ → (1 / (𝐶↑_𝑐1)) = (1 / 𝐶))
16	13, 15	eqtrd 2198	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → (𝐶↑_𝑐-1) = (1 / 𝐶))
17	16	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶↑_𝑐-1) = (1 / 𝐶))
18	17	oveq2d 5858	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · (𝐶↑_𝑐-1)) = (𝐴 · (1 / 𝐶)))
19	10, 18	eqtr4d 2201	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 / 𝐶) = (𝐴 · (𝐶↑_𝑐-1)))
20	19	adantl 275	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐴 / 𝐶) = (𝐴 · (𝐶↑_𝑐-1)))
21	20	oveq2d 5858	. 2 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 / 𝐶)) = (𝐵 log_b (𝐴 · (𝐶↑_𝑐-1))))
22		simpll 519	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐵 ∈ ℝ⁺)
23		simplr 520	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐵 # 1)
24		simprr 522	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐶 ∈ ℝ⁺)
25		rplogbcl 13504	. . . . 5 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ 𝐶 ∈ ℝ⁺) → (𝐵 log_b 𝐶) ∈ ℝ)
26	22, 23, 24, 25	syl3anc 1228	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) ∈ ℝ)
27		recn 7886	. . . 4 ⊢ ((𝐵 log_b 𝐶) ∈ ℝ → (𝐵 log_b 𝐶) ∈ ℂ)
28		mulm1 8298	. . . . 5 ⊢ ((𝐵 log_b 𝐶) ∈ ℂ → (-1 · (𝐵 log_b 𝐶)) = -(𝐵 log_b 𝐶))
29	28	oveq2d 5858	. . . 4 ⊢ ((𝐵 log_b 𝐶) ∈ ℂ → ((𝐵 log_b 𝐴) + (-1 · (𝐵 log_b 𝐶))) = ((𝐵 log_b 𝐴) + -(𝐵 log_b 𝐶)))
30	26, 27, 29	3syl 17	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((𝐵 log_b 𝐴) + (-1 · (𝐵 log_b 𝐶))) = ((𝐵 log_b 𝐴) + -(𝐵 log_b 𝐶)))
31		simprl 521	. . . . . 6 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐴 ∈ ℝ⁺)
32		rplogbcl 13504	. . . . . 6 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ 𝐴 ∈ ℝ⁺) → (𝐵 log_b 𝐴) ∈ ℝ)
33	22, 23, 31, 32	syl3anc 1228	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) ∈ ℝ)
34	33	recnd 7927	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) ∈ ℂ)
35	26	recnd 7927	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) ∈ ℂ)
36	34, 35	negsubd 8215	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((𝐵 log_b 𝐴) + -(𝐵 log_b 𝐶)) = ((𝐵 log_b 𝐴) − (𝐵 log_b 𝐶)))
37	30, 36	eqtr2d 2199	. 2 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((𝐵 log_b 𝐴) − (𝐵 log_b 𝐶)) = ((𝐵 log_b 𝐴) + (-1 · (𝐵 log_b 𝐶))))
38	3, 21, 37	3eqtr4d 2208	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 1343 ∈ wcel 2136 class class class wbr 3982 (class class class)co 5842 ℂcc 7751 ℝcr 7752 0cc0 7753 1c1 7754 + caddc 7756 · cmul 7758 − cmin 8069 -cneg 8070 # cap 8479 / cdiv 8568 ℝ⁺crp 9589 ↑_𝑐ccxp 13418 log_b clogb 13501

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 ax-pre-suploc 7874 ax-addf 7875 ax-mulf 7876

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-disj 3960 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-of 6050 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-map 6616 df-pm 6617 df-en 6707 df-dom 6708 df-fin 6709 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-ioo 9828 df-ico 9830 df-icc 9831 df-fz 9945 df-fzo 10078 df-seqfrec 10381 df-exp 10455 df-fac 10639 df-bc 10661 df-ihash 10689 df-shft 10757 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-clim 11220 df-sumdc 11295 df-ef 11589 df-e 11590 df-rest 12558 df-topgen 12577 df-psmet 12627 df-xmet 12628 df-met 12629 df-bl 12630 df-mopn 12631 df-top 12636 df-topon 12649 df-bases 12681 df-ntr 12736 df-cn 12828 df-cnp 12829 df-tx 12893 df-cncf 13198 df-limced 13265 df-dvap 13266 df-relog 13419 df-rpcxp 13420 df-logb 13502

This theorem is referenced by: logbrec 13518

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