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Theorem rprelogbmul 13523
Description: The logarithm of the product of two positive real numbers is the sum of logarithms. Property 2 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 29-May-2020.)
Assertion
Ref Expression
rprelogbmul |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A x. C ) ) = ( ( B logb A ) + ( B logb C ) ) )
Proof of Theorem rprelogbmul
StepHypRef Expression
1 relogmul 13440 . . . . 5 |- ( ( A e. RR+ /\ C e. RR+ ) -> ( log ` ( A x. C ) ) = ( ( log ` A ) + ( log ` C ) ) )
21adantl 275 . . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( log ` ( A x. C ) ) = ( ( log ` A ) + ( log ` C ) ) )
32oveq1d 5857 . . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( ( log ` ( A x. C ) ) / ( log ` B ) ) = ( ( ( log ` A ) + ( log ` C ) ) / ( log ` B ) ) )
4 relogcl 13433 . . . . . 6 |- ( A e. RR+ -> ( log ` A ) e. RR )
54recnd 7927 . . . . 5 |- ( A e. RR+ -> ( log ` A ) e. CC )
65ad2antrl 482 . . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( log ` A ) e. CC )
7 relogcl 13433 . . . . . 6 |- ( C e. RR+ -> ( log ` C ) e. RR )
87recnd 7927 . . . . 5 |- ( C e. RR+ -> ( log ` C ) e. CC )
98ad2antll 483 . . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( log ` C ) e. CC )
10 simpll 519 . . . . . 6 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> B e. RR+ )
1110relogcld 13453 . . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( log ` B ) e. RR )
1211recnd 7927 . . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( log ` B ) e. CC )
13 simplr 520 . . . . 5 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> B # 1 )
1410, 13logrpap0d 13449 . . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( log ` B ) # 0 )
156, 9, 12, 14divdirapd 8725 . . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( ( ( log ` A ) + ( log ` C ) ) / ( log ` B ) ) = ( ( ( log ` A ) / ( log ` B ) ) + ( ( log ` C ) / ( log ` B ) ) ) )
163, 15eqtrd 2198 . 2 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( ( log ` ( A x. C ) ) / ( log ` B ) ) = ( ( ( log ` A ) / ( log ` B ) ) + ( ( log ` C ) / ( log ` B ) ) ) )
17 rpmulcl 9614 . . . 4 |- ( ( A e. RR+ /\ C e. RR+ ) -> ( A x. C ) e. RR+ )
1817adantl 275 . . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( A x. C ) e. RR+ )
19 rplogbval 13513 . . 3 |- ( ( B e. RR+ /\ B # 1 /\ ( A x. C ) e. RR+ ) -> ( B logb ( A x. C ) ) = ( ( log ` ( A x. C ) ) / ( log ` B ) ) )
2010, 13, 18, 19syl3anc 1228 . 2 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A x. C ) ) = ( ( log ` ( A x. C ) ) / ( log ` B ) ) )
21 simprl 521 . . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> A e. RR+ )
22 rplogbval 13513 . . . 4 |- ( ( B e. RR+ /\ B # 1 /\ A e. RR+ ) -> ( B logb A ) = ( ( log ` A ) / ( log ` B ) ) )
2310, 13, 21, 22syl3anc 1228 . . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb A ) = ( ( log ` A ) / ( log ` B ) ) )
24 simprr 522 . . . 4 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> C e. RR+ )
25 rplogbval 13513 . . . 4 |- ( ( B e. RR+ /\ B # 1 /\ C e. RR+ ) -> ( B logb C ) = ( ( log ` C ) / ( log ` B ) ) )
2610, 13, 24, 25syl3anc 1228 . . 3 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb C ) = ( ( log ` C ) / ( log ` B ) ) )
2723, 26oveq12d 5860 . 2 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( ( B logb A ) + ( B logb C ) ) = ( ( ( log ` A ) / ( log ` B ) ) + ( ( log ` C ) / ( log ` B ) ) ) )
2816, 20, 273eqtr4d 2208 1 |- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A x. C ) ) = ( ( B logb A ) + ( B logb C ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   1c1 7754   + caddc 7756   x. cmul 7758   # cap 8479   / cdiv 8568   RR+crp 9589   logclog 13427   logb clogb 13511
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-pre-suploc 7874  ax-addf 7875  ax-mulf 7876
This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-map 6616  df-pm 6617  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-ioo 9828  df-ico 9830  df-icc 9831  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-shft 10757  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589  df-e 11590  df-rest 12558  df-topgen 12577  df-psmet 12637  df-xmet 12638  df-met 12639  df-bl 12640  df-mopn 12641  df-top 12646  df-topon 12659  df-bases 12691  df-ntr 12746  df-cn 12838  df-cnp 12839  df-tx 12903  df-cncf 13208  df-limced 13275  df-dvap 13276  df-relog 13429  df-logb 13512
This theorem is referenced by:  rprelogbmulexp  13524
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