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Theorem rplogbchbase 13278
Description: Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.)
Assertion
Ref Expression
rplogbchbase  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( A logb 
X )  =  ( ( B logb  X )  / 
( B logb  A ) ) )

Proof of Theorem rplogbchbase
StepHypRef Expression
1 simp3 984 . . . . 5  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  X  e.  RR+ )
21relogcld 13214 . . . 4  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( log `  X )  e.  RR )
32recnd 7906 . . 3  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( log `  X )  e.  CC )
4 simp1l 1006 . . . . 5  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  A  e.  RR+ )
54relogcld 13214 . . . 4  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( log `  A )  e.  RR )
65recnd 7906 . . 3  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( log `  A )  e.  CC )
7 simp2l 1008 . . . . 5  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  B  e.  RR+ )
87relogcld 13214 . . . 4  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( log `  B )  e.  RR )
98recnd 7906 . . 3  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( log `  B )  e.  CC )
10 simp1r 1007 . . . 4  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  A #  1 )
114, 10logrpap0d 13210 . . 3  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( log `  A ) #  0 )
12 simp2r 1009 . . . 4  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  B #  1 )
137, 12logrpap0d 13210 . . 3  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( log `  B ) #  0 )
143, 6, 9, 11, 13divcanap7d 8692 . 2  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  (
( ( log `  X
)  /  ( log `  B ) )  / 
( ( log `  A
)  /  ( log `  B ) ) )  =  ( ( log `  X )  /  ( log `  A ) ) )
15 rplogbval 13273 . . . 4  |-  ( ( B  e.  RR+  /\  B #  1  /\  X  e.  RR+ )  ->  ( B logb  X )  =  ( ( log `  X )  /  ( log `  B ) ) )
167, 12, 1, 15syl3anc 1220 . . 3  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( B logb 
X )  =  ( ( log `  X
)  /  ( log `  B ) ) )
17 rplogbval 13273 . . . 4  |-  ( ( B  e.  RR+  /\  B #  1  /\  A  e.  RR+ )  ->  ( B logb  A )  =  ( ( log `  A )  /  ( log `  B ) ) )
187, 12, 4, 17syl3anc 1220 . . 3  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( B logb 
A )  =  ( ( log `  A
)  /  ( log `  B ) ) )
1916, 18oveq12d 5842 . 2  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  (
( B logb  X )  / 
( B logb  A ) )  =  ( ( ( log `  X )  /  ( log `  B
) )  /  (
( log `  A
)  /  ( log `  B ) ) ) )
20 rplogbval 13273 . . 3  |-  ( ( A  e.  RR+  /\  A #  1  /\  X  e.  RR+ )  ->  ( A logb  X )  =  ( ( log `  X )  /  ( log `  A ) ) )
214, 10, 1, 20syl3anc 1220 . 2  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( A logb 
X )  =  ( ( log `  X
)  /  ( log `  A ) ) )
2214, 19, 213eqtr4rd 2201 1  |-  ( ( ( A  e.  RR+  /\  A #  1 )  /\  ( B  e.  RR+  /\  B #  1 )  /\  X  e.  RR+ )  ->  ( A logb 
X )  =  ( ( B logb  X )  / 
( B logb  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1335    e. wcel 2128   class class class wbr 3965   ` cfv 5170  (class class class)co 5824   1c1 7733   # cap 8456    / cdiv 8545   RR+crp 9560   logclog 13188   logb clogb 13271
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 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851  ax-caucvg 7852  ax-pre-suploc 7853  ax-addf 7854  ax-mulf 7855
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 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-isom 5179  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-of 6032  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-frec 6338  df-1o 6363  df-oadd 6367  df-er 6480  df-map 6595  df-pm 6596  df-en 6686  df-dom 6687  df-fin 6688  df-sup 6928  df-inf 6929  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-3 8893  df-4 8894  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-xneg 9679  df-xadd 9680  df-ioo 9796  df-ico 9798  df-icc 9799  df-fz 9913  df-fzo 10042  df-seqfrec 10345  df-exp 10419  df-fac 10600  df-bc 10622  df-ihash 10650  df-shft 10715  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-clim 11176  df-sumdc 11251  df-ef 11545  df-e 11546  df-rest 12364  df-topgen 12383  df-psmet 12398  df-xmet 12399  df-met 12400  df-bl 12401  df-mopn 12402  df-top 12407  df-topon 12420  df-bases 12452  df-ntr 12507  df-cn 12599  df-cnp 12600  df-tx 12664  df-cncf 12969  df-limced 13036  df-dvap 13037  df-relog 13190  df-logb 13272
This theorem is referenced by: (None)
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