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Theorem logbval 24421
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 11283. (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 5757 . . 3  |-  ( x  =  B  ->  ( log `  x )  =  ( log `  B
) )
21oveq2d 6126 . 2  |-  ( x  =  B  ->  (
( log `  y
)  /  ( log `  x ) )  =  ( ( log `  y
)  /  ( log `  B ) ) )
3 fveq2 5757 . . 3  |-  ( y  =  X  ->  ( log `  y )  =  ( log `  X
) )
43oveq1d 6125 . 2  |-  ( y  =  X  ->  (
( log `  y
)  /  ( log `  B ) )  =  ( ( log `  X
)  /  ( log `  B ) ) )
5 df-logb 24420 . 2  |- logb  =  ( x  e.  ( CC 
\  { 0 ,  1 } ) ,  y  e.  ( CC 
\  { 0 } )  |->  ( ( log `  y )  /  ( log `  x ) ) )
6 ovex 6135 . 2  |-  ( ( log `  X )  /  ( log `  B
) )  e.  _V
72, 4, 5, 6ovmpt2 6238 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 360    = wceq 1653    e. wcel 1727    \ cdif 3303   {csn 3838   {cpr 3839   ` cfv 5483  (class class class)co 6110   CCcc 9019   0cc0 9021   1c1 9022    / cdiv 9708   logclog 20483  logbclogb 24419
This theorem is referenced by:  logb2aval  24422  logbcl  24428  logbid1  24429  rnlogbval  24431  relogbcl  24433  logb1  24434  nnlogbexp  24435  elogb  28630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-logb 24420
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