Theorem rprelogbdiv 15671

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 9239	. . 3 |- -u 1 e. RR
2		rprelogbmulexp 15670	. . 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 1370	. 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 9887	. . . . . . 7 |- ( A e. RR+ -> A e. CC )
5	4	adantr 276	. . . . . 6 |- ( ( A e. RR+ /\ C e. RR+ ) -> A e. CC )
6		rpcn 9887	. . . . . . 7 |- ( C e. RR+ -> C e. CC )
7	6	adantl 277	. . . . . 6 |- ( ( A e. RR+ /\ C e. RR+ ) -> C e. CC )
8		rpap0 9895	. . . . . . 7 |- ( C e. RR+ -> C # 0 )
9	8	adantl 277	. . . . . 6 |- ( ( A e. RR+ /\ C e. RR+ ) -> C # 0 )
10	5, 7, 9	divrecapd 8963	. . . . 5 |- ( ( A e. RR+ /\ C e. RR+ ) -> ( A / C ) = ( A x. ( 1 / C ) ) )
11		ax-1cn 8115	. . . . . . . . 9 |- 1 e. CC
12		rpcxpneg 15621	. . . . . . . . 9 |- ( ( C e. RR+ /\ 1 e. CC ) -> ( C ^c -u 1 ) = ( 1 / ( C ^c 1 ) ) )
13	11, 12	mpan2 425	. . . . . . . 8 |- ( C e. RR+ -> ( C ^c -u 1 ) = ( 1 / ( C ^c 1 ) ) )
14		rpcxp1 15613	. . . . . . . . 9 |- ( C e. RR+ -> ( C ^c 1 ) = C )
15	14	oveq2d 6029	. . . . . . . 8 |- ( C e. RR+ -> ( 1 / ( C ^c 1 ) ) = ( 1 / C ) )
16	13, 15	eqtrd 2262	. . . . . . 7 |- ( C e. RR+ -> ( C ^c -u 1 ) = ( 1 / C ) )
17	16	adantl 277	. . . . . 6 |- ( ( A e. RR+ /\ C e. RR+ ) -> ( C ^c -u 1 ) = ( 1 / C ) )
18	17	oveq2d 6029	. . . . 5 |- ( ( A e. RR+ /\ C e. RR+ ) -> ( A x. ( C ^c -u 1 ) ) = ( A x. ( 1 / C ) ) )
19	10, 18	eqtr4d 2265	. . . 4 |- ( ( A e. RR+ /\ C e. RR+ ) -> ( A / C ) = ( A x. ( C ^c -u 1 ) ) )
20	19	adantl 277	. . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( A / C ) = ( A x. ( C ^c -u 1 ) ) )
21	20	oveq2d 6029	. 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 527	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> B e. RR+ )
23		simplr 528	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> B # 1 )
24		simprr 531	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> C e. RR+ )
25		rplogbcl 15660	. . . . 5 |- ( ( B e. RR+ /\ B # 1 /\ C e. RR+ ) -> ( B logb C ) e. RR )
26	22, 23, 24, 25	syl3anc 1271	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb C ) e. RR )
27		recn 8155	. . . 4 |- ( ( B logb C ) e. RR -> ( B logb C ) e. CC )
28		mulm1 8569	. . . . 5 |- ( ( B logb C ) e. CC -> ( -u 1 x. ( B logb C ) ) = -u ( B logb C ) )
29	28	oveq2d 6029	. . . 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 529	. . . . . 6 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> A e. RR+ )
32		rplogbcl 15660	. . . . . 6 |- ( ( B e. RR+ /\ B # 1 /\ A e. RR+ ) -> ( B logb A ) e. RR )
33	22, 23, 31, 32	syl3anc 1271	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb A ) e. RR )
34	33	recnd 8198	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb A ) e. CC )
35	26	recnd 8198	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb C ) e. CC )
36	34, 35	negsubd 8486	. . 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 2263	. 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 2272	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 104   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   CCcc 8020   RRcr 8021   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   - cmin 8340   -ucneg 8341   # cap 8751   / cdiv 8842   RR+crp 9878   ^c ccxp 15571   log_b clogb 15657

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142  ax-pre-suploc 8143  ax-addf 8144  ax-mulf 8145

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-disj 4063  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-oadd 6581  df-er 6697  df-map 6814  df-pm 6815  df-en 6905  df-dom 6906  df-fin 6907  df-sup 7174  df-inf 7175  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-xneg 9997  df-xadd 9998  df-ioo 10117  df-ico 10119  df-icc 10120  df-fz 10234  df-fzo 10368  df-seqfrec 10700  df-exp 10791  df-fac 10978  df-bc 11000  df-ihash 11028  df-shft 11366  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830  df-sumdc 11905  df-ef 12199  df-e 12200  df-rest 13314  df-topgen 13333  df-psmet 14547  df-xmet 14548  df-met 14549  df-bl 14550  df-mopn 14551  df-top 14712  df-topon 14725  df-bases 14757  df-ntr 14810  df-cn 14902  df-cnp 14903  df-tx 14967  df-cncf 15285  df-limced 15370  df-dvap 15371  df-relog 15572  df-rpcxp 15573  df-logb 15658

This theorem is referenced by:  logbrec  15674