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.

Step	Hyp	Ref	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
3	2	a1i 11	. . . 4 |- ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ ( x e. ( CC \ { 0 , 1 } ) /\ y e. ( CC \ { 0 } ) ) ) -> ( ( log ` y ) / ( log ` x ) ) e. _V )
4	3	ralrimivva 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 ) )
8	5, 7	ax-mp 5	. . . . . . 7 |- ( 1 e. { 0 } <-> 1 = 0 )
9	6, 8	nemtbir 2730	. . . . . 6 |- -. 1 e. { 0 }
10		eldif 3425	. . . . . 6 |- ( 1 e. ( CC \ { 0 } ) <-> ( 1 e. CC /\ -. 1 e. { 0 } ) )
11	5, 9, 10	mpbir2an 936	. . . . 5 |- 1 e. ( CC \ { 0 } )
12	11	ne0ii 3749	. . . 4 |- ( CC \ { 0 } ) =/= (/)
13	12	a1i 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 )
16	14, 15	ax-mp 5	. . . 4 |- ( CC \ { 0 } ) e. _V
17	16	a1i 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 ) )
19	18	biimpri 211	. . 3 |- ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) -> B e. ( CC \ { 0 , 1 } ) )
20	1, 4, 13, 17, 19	mpt2curryvald 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 )
23	22	a1i 11	. . . . . 6 |- ( B e. CC -> [_ B / x ]_ ( log ` x ) = ( log ` B ) )
24	23	oveq2d 6330	. . . . 5 |- ( B e. CC -> ( ( log ` y ) / [_ B / x ]_ ( log ` x ) ) = ( ( log ` y ) / ( log ` B ) ) )
25	21, 24	eqtrd 2495	. . . 4 |- ( B e. CC -> [_ B / x ]_ ( ( log ` y ) / ( log ` x ) ) = ( ( log ` y ) / ( log ` B ) ) )
26	25	3ad2ant1 1035	. . 3 |- ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) -> [_ B / x ]_ ( ( log ` y ) / ( log ` x ) ) = ( ( log ` y ) / ( log ` B ) ) )
27	26	mpteq2dv 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 ) ) ) )
28	20, 27	eqtrd 2495	1 |- ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) -> (curry logb ` B ) = ( y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` B ) ) ) )

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   log_b 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