Theorem rprelogbdiv 13234

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 e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A / C ) ) = ( ( B logb A ) - ( B logb C ) ) )

Proof of Theorem rprelogbdiv

Step	Hyp	Ref	Expression
1		neg1rr 8922	. . 3 |- -u 1 e. RR
2		rprelogbmulexp 13233	. . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ /\ -u 1 e. RR ) ) -> ( B logb ( A x. ( C ^c -u 1 ) ) ) = ( ( B logb A ) + ( -u 1 x. ( B logb C ) ) ) )
3	1, 2	mp3anr3 1318	. 2 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A x. ( C ^c -u 1 ) ) ) = ( ( B logb A ) + ( -u 1 x. ( B logb C ) ) ) )
4		rpcn 9551	. . . . . . 7 |- ( A e. RR+ -> A e. CC )
5	4	adantr 274	. . . . . 6 |- ( ( A e. RR+ /\ C e. RR+ ) -> A e. CC )
6		rpcn 9551	. . . . . . 7 |- ( C e. RR+ -> C e. CC )
7	6	adantl 275	. . . . . 6 |- ( ( A e. RR+ /\ C e. RR+ ) -> C e. CC )
8		rpap0 9559	. . . . . . 7 |- ( C e. RR+ -> C # 0 )
9	8	adantl 275	. . . . . 6 |- ( ( A e. RR+ /\ C e. RR+ ) -> C # 0 )
10	5, 7, 9	divrecapd 8649	. . . . 5 |- ( ( A e. RR+ /\ C e. RR+ ) -> ( A / C ) = ( A x. ( 1 / C ) ) )
11		ax-1cn 7808	. . . . . . . . 9 |- 1 e. CC
12		rpcxpneg 13188	. . . . . . . . 9 |- ( ( C e. RR+ /\ 1 e. CC ) -> ( C ^c -u 1 ) = ( 1 / ( C ^c 1 ) ) )
13	11, 12	mpan2 422	. . . . . . . 8 |- ( C e. RR+ -> ( C ^c -u 1 ) = ( 1 / ( C ^c 1 ) ) )
14		rpcxp1 13180	. . . . . . . . 9 |- ( C e. RR+ -> ( C ^c 1 ) = C )
15	14	oveq2d 5834	. . . . . . . 8 |- ( C e. RR+ -> ( 1 / ( C ^c 1 ) ) = ( 1 / C ) )
16	13, 15	eqtrd 2190	. . . . . . 7 |- ( C e. RR+ -> ( C ^c -u 1 ) = ( 1 / C ) )
17	16	adantl 275	. . . . . 6 |- ( ( A e. RR+ /\ C e. RR+ ) -> ( C ^c -u 1 ) = ( 1 / C ) )
18	17	oveq2d 5834	. . . . 5 |- ( ( A e. RR+ /\ C e. RR+ ) -> ( A x. ( C ^c -u 1 ) ) = ( A x. ( 1 / C ) ) )
19	10, 18	eqtr4d 2193	. . . 4 |- ( ( A e. RR+ /\ C e. RR+ ) -> ( A / C ) = ( A x. ( C ^c -u 1 ) ) )
20	19	adantl 275	. . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( A / C ) = ( A x. ( C ^c -u 1 ) ) )
21	20	oveq2d 5834	. 2 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A / C ) ) = ( B logb ( A x. ( C ^c -u 1 ) ) ) )
22		simpll 519	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> B e. RR+ )
23		simplr 520	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> B # 1 )
24		simprr 522	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> C e. RR+ )
25		rplogbcl 13223	. . . . 5 |- ( ( B e. RR+ /\ B # 1 /\ C e. RR+ ) -> ( B logb C ) e. RR )
26	22, 23, 24, 25	syl3anc 1220	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb C ) e. RR )
27		recn 7848	. . . 4 |- ( ( B logb C ) e. RR -> ( B logb C ) e. CC )
28		mulm1 8258	. . . . 5 |- ( ( B logb C ) e. CC -> ( -u 1 x. ( B logb C ) ) = -u ( B logb C ) )
29	28	oveq2d 5834	. . . 4 |- ( ( B logb C ) e. CC -> ( ( B logb A ) + ( -u 1 x. ( B logb C ) ) ) = ( ( B logb A ) + -u ( B logb C ) ) )
30	26, 27, 29	3syl 17	. . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( ( B logb A ) + ( -u 1 x. ( B logb C ) ) ) = ( ( B logb A ) + -u ( B logb C ) ) )
31		simprl 521	. . . . . 6 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> A e. RR+ )
32		rplogbcl 13223	. . . . . 6 |- ( ( B e. RR+ /\ B # 1 /\ A e. RR+ ) -> ( B logb A ) e. RR )
33	22, 23, 31, 32	syl3anc 1220	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb A ) e. RR )
34	33	recnd 7889	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb A ) e. CC )
35	26	recnd 7889	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb C ) e. CC )
36	34, 35	negsubd 8175	. . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( ( B logb A ) + -u ( B logb C ) ) = ( ( B logb A ) - ( B logb C ) ) )
37	30, 36	eqtr2d 2191	. 2 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( ( B logb A ) - ( B logb C ) ) = ( ( B logb A ) + ( -u 1 x. ( B logb C ) ) ) )
38	3, 21, 37	3eqtr4d 2200	1 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A / C ) ) = ( ( B logb A ) - ( B logb C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   class class class wbr 3965  (class class class)co 5818   CCcc 7713   RRcr 7714   0cc0 7715   1c1 7716   + caddc 7718   x. cmul 7720   - cmin 8029   -ucneg 8030   # cap 8439   / cdiv 8528   RR+crp 9542   ^c ccxp 13138   log_b clogb 13220

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 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835  ax-pre-suploc 7836  ax-addf 7837  ax-mulf 7838

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 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-of 6026  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-frec 6332  df-1o 6357  df-oadd 6361  df-er 6473  df-map 6588  df-pm 6589  df-en 6679  df-dom 6680  df-fin 6681  df-sup 6920  df-inf 6921  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-xneg 9661  df-xadd 9662  df-ioo 9778  df-ico 9780  df-icc 9781  df-fz 9895  df-fzo 10024  df-seqfrec 10327  df-exp 10401  df-fac 10582  df-bc 10604  df-ihash 10632  df-shft 10697  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881  df-clim 11158  df-sumdc 11233  df-ef 11527  df-e 11528  df-rest 12313  df-topgen 12332  df-psmet 12347  df-xmet 12348  df-met 12349  df-bl 12350  df-mopn 12351  df-top 12356  df-topon 12369  df-bases 12401  df-ntr 12456  df-cn 12548  df-cnp 12549  df-tx 12613  df-cncf 12918  df-limced 12985  df-dvap 12986  df-relog 13139  df-rpcxp 13140  df-logb 13221

This theorem is referenced by:  logbrec  13237