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Theorem logb2aval 24187
Description: Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the 2-argument form logb <. B ,  X >. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
Assertion
Ref Expression
logb2aval  |-  ( ( B  e.  ( CC 
\  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } ) )  ->  (logb `  <. B ,  X >. )  =  ( ( log `  X )  /  ( log `  B ) ) )

Proof of Theorem logb2aval
StepHypRef Expression
1 df-ov 6023 . 2  |-  ( Blogb X )  =  (logb `  <. B ,  X >. )
2 logbval 24186 . 2  |-  ( ( B  e.  ( CC 
\  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } ) )  ->  ( Blogb X )  =  ( ( log `  X
)  /  ( log `  B ) ) )
31, 2syl5eqr 2433 1  |-  ( ( B  e.  ( CC 
\  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } ) )  ->  (logb `  <. B ,  X >. )  =  ( ( log `  X )  /  ( log `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    \ cdif 3260   {csn 3757   {cpr 3758   <.cop 3760   ` cfv 5394  (class class class)co 6020   CCcc 8921   0cc0 8923   1c1 8924    / cdiv 9609   logclog 20319  logbclogb 24184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-logb 24185
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