rprelogbdiv - Intuitionistic Logic Explorer

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Theorem rprelogbdiv 13245

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 8933	. . 3 ⊢ -1 ∈ ℝ
2		rprelogbmulexp 13244	. . 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 1318	. 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 9562	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
5	4	adantr 274	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
6		rpcn 9562	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ∈ ℂ)
7	6	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
8		rpap0 9570	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 # 0)
9	8	adantl 275	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 # 0)
10	5, 7, 9	divrecapd 8660	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶)))
11		ax-1cn 7819	. . . . . . . . 9 ⊢ 1 ∈ ℂ
12		rpcxpneg 13199	. . . . . . . . 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 13191	. . . . . . . . 9 ⊢ (𝐶 ∈ ℝ⁺ → (𝐶↑_𝑐1) = 𝐶)
15	14	oveq2d 5837	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ⁺ → (1 / (𝐶↑_𝑐1)) = (1 / 𝐶))
16	13, 15	eqtrd 2190	. . . . . . 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 5837	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · (𝐶↑_𝑐-1)) = (𝐴 · (1 / 𝐶)))
19	10, 18	eqtr4d 2193	. . . 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 5837	. 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 13234	. . . . 5 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ 𝐶 ∈ ℝ⁺) → (𝐵 log_b 𝐶) ∈ ℝ)
26	22, 23, 24, 25	syl3anc 1220	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) ∈ ℝ)
27		recn 7859	. . . 4 ⊢ ((𝐵 log_b 𝐶) ∈ ℝ → (𝐵 log_b 𝐶) ∈ ℂ)
28		mulm1 8269	. . . . 5 ⊢ ((𝐵 log_b 𝐶) ∈ ℂ → (-1 · (𝐵 log_b 𝐶)) = -(𝐵 log_b 𝐶))
29	28	oveq2d 5837	. . . 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 13234	. . . . . 6 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ 𝐴 ∈ ℝ⁺) → (𝐵 log_b 𝐴) ∈ ℝ)
33	22, 23, 31, 32	syl3anc 1220	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) ∈ ℝ)
34	33	recnd 7900	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) ∈ ℂ)
35	26	recnd 7900	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) ∈ ℂ)
36	34, 35	negsubd 8186	. . 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 2191	. 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 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 (class class class)co 5821 ℂcc 7724 ℝcr 7725 0cc0 7726 1c1 7727 + caddc 7729 · cmul 7731 − cmin 8040 -cneg 8041 # cap 8450 / cdiv 8539 ℝ⁺crp 9553 ↑_𝑐ccxp 13149 log_b clogb 13231

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 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7817 ax-resscn 7818 ax-1cn 7819 ax-1re 7820 ax-icn 7821 ax-addcl 7822 ax-addrcl 7823 ax-mulcl 7824 ax-mulrcl 7825 ax-addcom 7826 ax-mulcom 7827 ax-addass 7828 ax-mulass 7829 ax-distr 7830 ax-i2m1 7831 ax-0lt1 7832 ax-1rid 7833 ax-0id 7834 ax-rnegex 7835 ax-precex 7836 ax-cnre 7837 ax-pre-ltirr 7838 ax-pre-ltwlin 7839 ax-pre-lttrn 7840 ax-pre-apti 7841 ax-pre-ltadd 7842 ax-pre-mulgt0 7843 ax-pre-mulext 7844 ax-arch 7845 ax-caucvg 7846 ax-pre-suploc 7847 ax-addf 7848 ax-mulf 7849

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 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-isom 5178 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-of 6029 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-frec 6335 df-1o 6360 df-oadd 6364 df-er 6477 df-map 6592 df-pm 6593 df-en 6683 df-dom 6684 df-fin 6685 df-sup 6924 df-inf 6925 df-pnf 7908 df-mnf 7909 df-xr 7910 df-ltxr 7911 df-le 7912 df-sub 8042 df-neg 8043 df-reap 8444 df-ap 8451 df-div 8540 df-inn 8828 df-2 8886 df-3 8887 df-4 8888 df-n0 9085 df-z 9162 df-uz 9434 df-q 9522 df-rp 9554 df-xneg 9672 df-xadd 9673 df-ioo 9789 df-ico 9791 df-icc 9792 df-fz 9906 df-fzo 10035 df-seqfrec 10338 df-exp 10412 df-fac 10593 df-bc 10615 df-ihash 10643 df-shft 10708 df-cj 10735 df-re 10736 df-im 10737 df-rsqrt 10891 df-abs 10892 df-clim 11169 df-sumdc 11244 df-ef 11538 df-e 11539 df-rest 12324 df-topgen 12343 df-psmet 12358 df-xmet 12359 df-met 12360 df-bl 12361 df-mopn 12362 df-top 12367 df-topon 12380 df-bases 12412 df-ntr 12467 df-cn 12559 df-cnp 12560 df-tx 12624 df-cncf 12929 df-limced 12996 df-dvap 12997 df-relog 13150 df-rpcxp 13151 df-logb 13232

This theorem is referenced by: logbrec 13248

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