relogbdiv - Metamath Proof Explorer

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Theorem relogbdiv 26999

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
relogbdiv	⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 / 𝐶)) = ((𝐵 log_b 𝐴) − (𝐵 log_b 𝐶)))

**Proof of Theorem relogbdiv**
Step	Hyp	Ref	Expression
1		neg1rr 12223	. . 3 ⊢ -1 ∈ ℝ
2		relogbmulexp 26998	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 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 1489	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 · (𝐶↑_𝑐-1))) = ((𝐵 log_b 𝐴) + (-1 · (𝐵 log_b 𝐶))))
4		rpcn 13047	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
5	4	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ ℂ)
6		rpcn 13047	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ∈ ℂ)
7	6	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
8		rpne0 13053	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ≠ 0)
9	8	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ≠ 0)
10	5, 7, 9	divrecd 12013	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶)))
11		1cnd 11221	. . . . . . . . 9 ⊢ (𝐶 ∈ ℝ⁺ → 1 ∈ ℂ)
12	6, 8, 11	cxpnegd 26935	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ⁺ → (𝐶↑_𝑐-1) = (1 / (𝐶↑_𝑐1)))
13	6	cxp1d 26926	. . . . . . . . 9 ⊢ (𝐶 ∈ ℝ⁺ → (𝐶↑_𝑐1) = 𝐶)
14	13	oveq2d 7435	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ⁺ → (1 / (𝐶↑_𝑐1)) = (1 / 𝐶))
15	12, 14	eqtrd 2800	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → (𝐶↑_𝑐-1) = (1 / 𝐶))
16	15	adantl 487	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐶↑_𝑐-1) = (1 / 𝐶))
17	16	oveq2d 7435	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 · (𝐶↑_𝑐-1)) = (𝐴 · (1 / 𝐶)))
18	10, 17	eqtr4d 2803	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → (𝐴 / 𝐶) = (𝐴 · (𝐶↑_𝑐-1)))
19	18	adantl 487	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐴 / 𝐶) = (𝐴 · (𝐶↑_𝑐-1)))
20	19	oveq2d 7435	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 / 𝐶)) = (𝐵 log_b (𝐴 · (𝐶↑_𝑐-1))))
21		rpcndif0 13057	. . . . . 6 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ∈ (ℂ ∖ {0}))
22	21	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐶 ∈ (ℂ ∖ {0}))
23		logbcl 26987	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐶 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝐶) ∈ ℂ)
24	22, 23	sylan2 605	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐶) ∈ ℂ)
25		mulm1 11674	. . . . 5 ⊢ ((𝐵 log_b 𝐶) ∈ ℂ → (-1 · (𝐵 log_b 𝐶)) = -(𝐵 log_b 𝐶))
26	25	oveq2d 7435	. . . 4 ⊢ ((𝐵 log_b 𝐶) ∈ ℂ → ((𝐵 log_b 𝐴) + (-1 · (𝐵 log_b 𝐶))) = ((𝐵 log_b 𝐴) + -(𝐵 log_b 𝐶)))
27	24, 26	syl 18	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((𝐵 log_b 𝐴) + (-1 · (𝐵 log_b 𝐶))) = ((𝐵 log_b 𝐴) + -(𝐵 log_b 𝐶)))
28		rpcndif0 13057	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ (ℂ ∖ {0}))
29	28	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺) → 𝐴 ∈ (ℂ ∖ {0}))
30		logbcl 26987	. . . . 5 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝐴 ∈ (ℂ ∖ {0})) → (𝐵 log_b 𝐴) ∈ ℂ)
31	29, 30	sylan2 605	. . . 4 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b 𝐴) ∈ ℂ)
32	31, 24	negsubd 11594	. . 3 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((𝐵 log_b 𝐴) + -(𝐵 log_b 𝐶)) = ((𝐵 log_b 𝐴) − (𝐵 log_b 𝐶)))
33	27, 32	eqtr2d 2801	. 2 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → ((𝐵 log_b 𝐴) − (𝐵 log_b 𝐶)) = ((𝐵 log_b 𝐴) + (-1 · (𝐵 log_b 𝐶))))
34	3, 20, 33	3eqtr4d 2810	1 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ (𝐴 ∈ ℝ⁺ ∧ 𝐶 ∈ ℝ⁺)) → (𝐵 log_b (𝐴 / 𝐶)) = ((𝐵 log_b 𝐴) − (𝐵 log_b 𝐶)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∖ cdif 3903 {csn 4591 {cpr 4593 (class class class)co 7419 ℂcc 11117 ℝcr 11118 0cc0 11119 1c1 11120 + caddc 11122 · cmul 11124 − cmin 11460 -cneg 11461 / cdiv 11890 ℝ⁺crp 13036 ↑_𝑐ccxp 26775 log_b clogb 26984

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196 ax-pre-sup 11197 ax-addf 11198

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-div 11891 df-nn 12253 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12524 df-z 12611 df-dec 12732 df-uz 12883 df-q 12993 df-rp 13037 df-xneg 13157 df-xadd 13158 df-xmul 13159 df-ioo 13396 df-ioc 13397 df-ico 13398 df-icc 13399 df-fz 13556 df-fzo 13704 df-fl 13847 df-mod 13925 df-seq 14060 df-exp 14120 df-fac 14332 df-bc 14361 df-hash 14389 df-shft 15132 df-cj 15178 df-re 15179 df-im 15180 df-sqrt 15314 df-abs 15315 df-limsup 15550 df-clim 15567 df-rlim 15568 df-sum 15766 df-ef 16147 df-sin 16149 df-cos 16150 df-pi 16152 df-struct 17233 df-sets 17250 df-slot 17268 df-ndx 17280 df-base 17296 df-ress 17317 df-plusg 17349 df-mulr 17350 df-starv 17351 df-sca 17352 df-vsca 17353 df-ip 17354 df-tset 17355 df-ple 17356 df-ds 17358 df-unif 17359 df-hom 17360 df-cco 17361 df-rest 17501 df-topn 17502 df-0g 17520 df-gsum 17521 df-topgen 17522 df-pt 17523 df-prds 17526 df-xrs 17582 df-qtop 17587 df-imas 17588 df-xps 17590 df-mre 17664 df-mrc 17665 df-acs 17667 df-mgm 18724 df-sgrp 18813 df-mnd 18829 df-submnd 18883 df-mulg 19182 df-cntz 19435 df-cmn 19900 df-psmet 21568 df-xmet 21569 df-met 21570 df-bl 21571 df-mopn 21572 df-fbas 21573 df-fg 21574 df-cnfld 21577 df-top 23105 df-topon 23122 df-topsp 23144 df-bases 23157 df-cld 23230 df-ntr 23231 df-cls 23232 df-nei 23309 df-lp 23347 df-perf 23348 df-cn 23438 df-cnp 23439 df-haus 23526 df-tx 23774 df-hmeo 23967 df-fil 24058 df-fm 24150 df-flim 24151 df-flf 24152 df-xms 24532 df-ms 24533 df-tms 24534 df-cncf 25092 df-limc 26080 df-dv 26081 df-log 26776 df-cxp 26777 df-logb 26985

This theorem is used by: relogbdivb 49418

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