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Theorem logb2aval 24383
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 6076 . 2  |-  ( Blogb X )  =  (logb `  <. B ,  X >. )
2 logbval 24382 . 2  |-  ( ( B  e.  ( CC 
\  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } ) )  ->  ( Blogb X )  =  ( ( log `  X
)  /  ( log `  B ) ) )
31, 2syl5eqr 2481 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 1652    e. wcel 1725    \ cdif 3309   {csn 3806   {cpr 3807   <.cop 3809   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    / cdiv 9669   logclog 20444  logbclogb 24380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-logb 24381
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