Theorem rplogbreexp 16055

Description: Power law for the general logarithm for real powers: The logarithm of a positive real number to the power of a real number is equal to the product of the exponent and the logarithm of the base of the power. Property 4 of [Cohen4] p. 361. (Contributed by AV, 9-Jun-2020.)

Assertion

Ref	Expression
rplogbreexp	|- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( B logb ( C ^c E ) ) = ( E x. ( B logb C ) ) )

Proof of Theorem rplogbreexp

Step	Hyp	Ref	Expression
1		logcxp 15999	. . . . 5 |- ( ( C e. RR+ /\ E e. RR ) -> ( log ` ( C ^c E ) ) = ( E x. ( log ` C ) ) )
2	1	3adant1 1046	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( log ` ( C ^c E ) ) = ( E x. ( log ` C ) ) )
3	2	oveq1d 6100	. . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( ( log ` ( C ^c E ) ) / ( log ` B ) ) = ( ( E x. ( log ` C ) ) / ( log ` B ) ) )
4		simp3 1030	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> E e. RR )
5	4	recnd 8354	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> E e. CC )
6		simp2 1029	. . . . . 6 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> C e. RR+ )
7	6	relogcld 15983	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( log ` C ) e. RR )
8	7	recnd 8354	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( log ` C ) e. CC )
9		simp1l 1052	. . . . . 6 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> B e. RR+ )
10	9	relogcld 15983	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( log ` B ) e. RR )
11	10	recnd 8354	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( log ` B ) e. CC )
12		simp1r 1053	. . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> B # 1 )
13	9, 12	logrpap0d 15979	. . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( log ` B ) # 0 )
14	5, 8, 11, 13	divassapd 9156	. . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( ( E x. ( log ` C ) ) / ( log ` B ) ) = ( E x. ( ( log ` C ) / ( log ` B ) ) ) )
15	3, 14	eqtrd 2271	. 2 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( ( log ` ( C ^c E ) ) / ( log ` B ) ) = ( E x. ( ( log ` C ) / ( log ` B ) ) ) )
16	6, 4	rpcxpcld 16035	. . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( C ^c E ) e. RR+ )
17		rplogbval 16047	. . 3 |- ( ( B e. RR+ /\ B # 1 /\ ( C ^c E ) e. RR+ ) -> ( B logb ( C ^c E ) ) = ( ( log ` ( C ^c E ) ) / ( log ` B ) ) )
18	9, 12, 16, 17	syl3anc 1278	. 2 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( B logb ( C ^c E ) ) = ( ( log ` ( C ^c E ) ) / ( log ` B ) ) )
19		rplogbval 16047	. . . 4 |- ( ( B e. RR+ /\ B # 1 /\ C e. RR+ ) -> ( B logb C ) = ( ( log ` C ) / ( log ` B ) ) )
20	9, 12, 6, 19	syl3anc 1278	. . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( B logb C ) = ( ( log ` C ) / ( log ` B ) ) )
21	20	oveq2d 6101	. 2 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( E x. ( B logb C ) ) = ( E x. ( ( log ` C ) / ( log ` B ) ) ) )
22	15, 18, 21	3eqtr4d 2281	1 |- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( B logb ( C ^c E ) ) = ( E x. ( B logb C ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   RRcr 8178   1c1 8180   x. cmul 8184   # cap 8909   / cdiv 9002   RR+crp 10054   logclog 15957   ^c ccxp 15958   log_b clogb 16045

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-ioo 10294  df-ico 10296  df-icc 10297  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-bc 11186  df-ihash 11215  df-shft 11580  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120  df-ef 12415  df-e 12416  df-rest 13595  df-topgen 13614  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883  df-mopn 14884  df-top 15099  df-topon 15112  df-bases 15144  df-ntr 15197  df-cn 15289  df-cnp 15290  df-tx 15354  df-cncf 15672  df-limced 15757  df-dvap 15758  df-relog 15959  df-rpcxp 15960  df-logb 16046

This theorem is used by:  rplogbzexp  16056  rprelogbmulexp  16058  rplogbcxp  16065