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Theorem logbval 24351
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 11215. (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 5695 . . 3  |-  ( x  =  B  ->  ( log `  x )  =  ( log `  B
) )
21oveq2d 6064 . 2  |-  ( x  =  B  ->  (
( log `  y
)  /  ( log `  x ) )  =  ( ( log `  y
)  /  ( log `  B ) ) )
3 fveq2 5695 . . 3  |-  ( y  =  X  ->  ( log `  y )  =  ( log `  X
) )
43oveq1d 6063 . 2  |-  ( y  =  X  ->  (
( log `  y
)  /  ( log `  B ) )  =  ( ( log `  X
)  /  ( log `  B ) ) )
5 df-logb 24350 . 2  |- logb  =  ( x  e.  ( CC 
\  { 0 ,  1 } ) ,  y  e.  ( CC 
\  { 0 } )  |->  ( ( log `  y )  /  ( log `  x ) ) )
6 ovex 6073 . 2  |-  ( ( log `  X )  /  ( log `  B
) )  e.  _V
72, 4, 5, 6ovmpt2 6176 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 359    = wceq 1649    e. wcel 1721    \ cdif 3285   {csn 3782   {cpr 3783   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954   1c1 8955    / cdiv 9641   logclog 20413  logbclogb 24349
This theorem is referenced by:  logb2aval  24352  logbcl  24358  logbid1  24359  rnlogbval  24361  relogbcl  24363  logb1  24364  nnlogbexp  24365  elogb  28254
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-logb 24350
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