rprelogbdiv - Intuitionistic Logic Explorer

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Theorem rprelogbdiv 15089

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 9088	. . 3 ⊢ -1 ∈ ℝ
2		rprelogbmulexp 15088	. . 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 1347	. 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 9728	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
5	4	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
6		rpcn 9728	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ∈ ℂ)
7	6	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
8		rpap0 9736	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 # 0)
9	8	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 # 0)
10	5, 7, 9	divrecapd 8812	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶)))
11		ax-1cn 7965	. . . . . . . . 9 ⊢ 1 ∈ ℂ
12		rpcxpneg 15042	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 1 ∈ ℂ) → (𝐶↑_𝑐-1) = (1 / (𝐶↑_𝑐1)))
13	11, 12	mpan2 425	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ⁺ → (𝐶↑_𝑐-1) = (1 / (𝐶↑_𝑐1)))
14		rpcxp1 15034	. . . . . . . . 9 ⊢ (𝐶 ∈ ℝ⁺ → (𝐶↑_𝑐1) = 𝐶)
15	14	oveq2d 5934	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ⁺ → (1 / (𝐶↑_𝑐1)) = (1 / 𝐶))
16	13, 15	eqtrd 2226	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → (𝐶↑_𝑐-1) = (1 / 𝐶))
17	16	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶↑_𝑐-1) = (1 / 𝐶))
18	17	oveq2d 5934	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · (𝐶↑_𝑐-1)) = (𝐴 · (1 / 𝐶)))
19	10, 18	eqtr4d 2229	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 / 𝐶) = (𝐴 · (𝐶↑_𝑐-1)))
20	19	adantl 277	. . 3 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐴 / 𝐶) = (𝐴 · (𝐶↑_𝑐-1)))
21	20	oveq2d 5934	. 2 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 / 𝐶)) = (𝐵 log_b (𝐴 · (𝐶↑_𝑐-1))))
22		simpll 527	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐵 ∈ ℝ⁺)
23		simplr 528	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐵 # 1)
24		simprr 531	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐶 ∈ ℝ⁺)
25		rplogbcl 15078	. . . . 5 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ 𝐶 ∈ ℝ⁺) → (𝐵 log_b 𝐶) ∈ ℝ)
26	22, 23, 24, 25	syl3anc 1249	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) ∈ ℝ)
27		recn 8005	. . . 4 ⊢ ((𝐵 log_b 𝐶) ∈ ℝ → (𝐵 log_b 𝐶) ∈ ℂ)
28		mulm1 8419	. . . . 5 ⊢ ((𝐵 log_b 𝐶) ∈ ℂ → (-1 · (𝐵 log_b 𝐶)) = -(𝐵 log_b 𝐶))
29	28	oveq2d 5934	. . . 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 529	. . . . . 6 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → 𝐴 ∈ ℝ⁺)
32		rplogbcl 15078	. . . . . 6 ⊢ ((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1 ∧ 𝐴 ∈ ℝ⁺) → (𝐵 log_b 𝐴) ∈ ℝ)
33	22, 23, 31, 32	syl3anc 1249	. . . . 5 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) ∈ ℝ)
34	33	recnd 8048	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) ∈ ℂ)
35	26	recnd 8048	. . . 4 ⊢ (((𝐵 ∈ ℝ⁺ ∧ 𝐵 # 1) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) ∈ ℂ)
36	34, 35	negsubd 8336	. . 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 2227	. 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 2236	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 1364 ∈ wcel 2164 class class class wbr 4029 (class class class)co 5918 ℂcc 7870 ℝcr 7871 0cc0 7872 1c1 7873 + caddc 7875 · cmul 7877 − cmin 8190 -cneg 8191 # cap 8600 / cdiv 8691 ℝ⁺crp 9719 ↑_𝑐ccxp 14992 log_b clogb 15075

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992 ax-pre-suploc 7993 ax-addf 7994 ax-mulf 7995

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-disj 4007 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-of 6130 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-map 6704 df-pm 6705 df-en 6795 df-dom 6796 df-fin 6797 df-sup 7043 df-inf 7044 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-xneg 9838 df-xadd 9839 df-ioo 9958 df-ico 9960 df-icc 9961 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-fac 10797 df-bc 10819 df-ihash 10847 df-shft 10959 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-sumdc 11497 df-ef 11791 df-e 11792 df-rest 12852 df-topgen 12871 df-psmet 14039 df-xmet 14040 df-met 14041 df-bl 14042 df-mopn 14043 df-top 14166 df-topon 14179 df-bases 14211 df-ntr 14264 df-cn 14356 df-cnp 14357 df-tx 14421 df-cncf 14726 df-limced 14810 df-dvap 14811 df-relog 14993 df-rpcxp 14994 df-logb 15076

This theorem is referenced by: logbrec 15092

W3C validator