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Theorem logbmpt 23773
Description: The general logarithm to a fixed base regarded as mapping. (Contributed by AV, 11-Jun-2020.)
Assertion
Ref Expression
logbmpt  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  (curry logb  `  B
)  =  ( y  e.  ( CC  \  { 0 } ) 
|->  ( ( log `  y
)  /  ( log `  B ) ) ) )
Distinct variable group:    y, B

Proof of Theorem logbmpt
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-logb 23750 . . 3  |- logb  =  (
x  e.  ( CC 
\  { 0 ,  1 } ) ,  y  e.  ( CC 
\  { 0 } )  |->  ( ( log `  y )  /  ( log `  x ) ) )
2 ovex 6342 . . . . 5  |-  ( ( log `  y )  /  ( log `  x
) )  e.  _V
32a1i 11 . . . 4  |-  ( ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  /\  ( x  e.  ( CC  \  {
0 ,  1 } )  /\  y  e.  ( CC  \  {
0 } ) ) )  ->  ( ( log `  y )  / 
( log `  x
) )  e.  _V )
43ralrimivva 2820 . . 3  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  A. x  e.  ( CC  \  {
0 ,  1 } ) A. y  e.  ( CC  \  {
0 } ) ( ( log `  y
)  /  ( log `  x ) )  e. 
_V )
5 ax-1cn 9622 . . . . . 6  |-  1  e.  CC
6 ax-1ne0 9633 . . . . . . 7  |-  1  =/=  0
7 elsncg 4002 . . . . . . . 8  |-  ( 1  e.  CC  ->  (
1  e.  { 0 }  <->  1  =  0 ) )
85, 7ax-mp 5 . . . . . . 7  |-  ( 1  e.  { 0 }  <->  1  =  0 )
96, 8nemtbir 2730 . . . . . 6  |-  -.  1  e.  { 0 }
10 eldif 3425 . . . . . 6  |-  ( 1  e.  ( CC  \  { 0 } )  <-> 
( 1  e.  CC  /\ 
-.  1  e.  {
0 } ) )
115, 9, 10mpbir2an 936 . . . . 5  |-  1  e.  ( CC  \  {
0 } )
1211ne0ii 3749 . . . 4  |-  ( CC 
\  { 0 } )  =/=  (/)
1312a1i 11 . . 3  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  ( CC  \  { 0 } )  =/=  (/) )
14 cnex 9645 . . . . 5  |-  CC  e.  _V
15 difexg 4564 . . . . 5  |-  ( CC  e.  _V  ->  ( CC  \  { 0 } )  e.  _V )
1614, 15ax-mp 5 . . . 4  |-  ( CC 
\  { 0 } )  e.  _V
1716a1i 11 . . 3  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  ( CC  \  { 0 } )  e.  _V )
18 eldifpr 4001 . . . 4  |-  ( B  e.  ( CC  \  { 0 ,  1 } )  <->  ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 ) )
1918biimpri 211 . . 3  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  B  e.  ( CC  \  {
0 ,  1 } ) )
201, 4, 13, 17, 19mpt2curryvald 7042 . 2  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  (curry logb  `  B
)  =  ( y  e.  ( CC  \  { 0 } ) 
|->  [_ B  /  x ]_ ( ( log `  y
)  /  ( log `  x ) ) ) )
21 csbov2g 6352 . . . . 5  |-  ( B  e.  CC  ->  [_ B  /  x ]_ ( ( log `  y )  /  ( log `  x
) )  =  ( ( log `  y
)  /  [_ B  /  x ]_ ( log `  x ) ) )
22 csbfv 5924 . . . . . . 7  |-  [_ B  /  x ]_ ( log `  x )  =  ( log `  B )
2322a1i 11 . . . . . 6  |-  ( B  e.  CC  ->  [_ B  /  x ]_ ( log `  x )  =  ( log `  B ) )
2423oveq2d 6330 . . . . 5  |-  ( B  e.  CC  ->  (
( log `  y
)  /  [_ B  /  x ]_ ( log `  x ) )  =  ( ( log `  y
)  /  ( log `  B ) ) )
2521, 24eqtrd 2495 . . . 4  |-  ( B  e.  CC  ->  [_ B  /  x ]_ ( ( log `  y )  /  ( log `  x
) )  =  ( ( log `  y
)  /  ( log `  B ) ) )
26253ad2ant1 1035 . . 3  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  [_ B  /  x ]_ ( ( log `  y )  /  ( log `  x
) )  =  ( ( log `  y
)  /  ( log `  B ) ) )
2726mpteq2dv 4503 . 2  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  (
y  e.  ( CC 
\  { 0 } )  |->  [_ B  /  x ]_ ( ( log `  y
)  /  ( log `  x ) ) )  =  ( y  e.  ( CC  \  {
0 } )  |->  ( ( log `  y
)  /  ( log `  B ) ) ) )
2820, 27eqtrd 2495 1  |-  ( ( B  e.  CC  /\  B  =/=  0  /\  B  =/=  1 )  ->  (curry logb  `  B
)  =  ( y  e.  ( CC  \  { 0 } ) 
|->  ( ( log `  y
)  /  ( log `  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   _Vcvv 3056   [_csb 3374    \ cdif 3412   (/)c0 3742   {csn 3979   {cpr 3981    |-> cmpt 4474   ` cfv 5600  (class class class)co 6314  curry ccur 7037   CCcc 9562   0cc0 9564   1c1 9565    / cdiv 10296   logclog 23552   logb clogb 23749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-1cn 9622  ax-1ne0 9633
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-1st 6819  df-2nd 6820  df-cur 7039  df-logb 23750
This theorem is referenced by:  logbf  23774  relogbf  23776  logblog  23777
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