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Theorem logbval 23656
Description: Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the other operand here. Proof is similar to modval 11067. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
Assertion
Ref Expression
logbval  |-  ( ( B  e.  ( CC 
\  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } ) )  ->  ( Blogb X )  =  ( ( log `  X
)  /  ( log `  B ) ) )

Proof of Theorem logbval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5608 . . 3  |-  ( x  =  B  ->  ( log `  x )  =  ( log `  B
) )
21oveq2d 5961 . 2  |-  ( x  =  B  ->  (
( log `  y
)  /  ( log `  x ) )  =  ( ( log `  y
)  /  ( log `  B ) ) )
3 fveq2 5608 . . 3  |-  ( y  =  X  ->  ( log `  y )  =  ( log `  X
) )
43oveq1d 5960 . 2  |-  ( y  =  X  ->  (
( log `  y
)  /  ( log `  B ) )  =  ( ( log `  X
)  /  ( log `  B ) ) )
5 df-logb 23655 . 2  |- logb  =  ( x  e.  ( CC 
\  { 0 ,  1 } ) ,  y  e.  ( CC 
\  { 0 } )  |->  ( ( log `  y )  /  ( log `  x ) ) )
6 ovex 5970 . 2  |-  ( ( log `  X )  /  ( log `  B
) )  e.  _V
72, 4, 5, 6ovmpt2 6070 1  |-  ( ( B  e.  ( CC 
\  { 0 ,  1 } )  /\  X  e.  ( CC  \  { 0 } ) )  ->  ( Blogb X )  =  ( ( log `  X
)  /  ( log `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    \ cdif 3225   {csn 3716   {cpr 3717   ` cfv 5337  (class class class)co 5945   CCcc 8825   0cc0 8827   1c1 8828    / cdiv 9513   logclog 20019  logbclogb 23654
This theorem is referenced by:  logb2aval  23657  logbcl  23663  logbid1  23664  rnlogbval  23666  relogbcl  23668  logb1  23669  nnlogbexp  23670  elogb  27959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-logb 23655
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