Theorem logbfval 24528

Description: The general logarithm of a complex number to a fixed base. (Contributed by AV, 11-Jun-2020.)

Assertion

Ref	Expression
logbfval	|- ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> ( (curry logb ` B ) ` X ) = ( B logb X ) )

Proof of Theorem logbfval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-logb 24503	. 2 |- logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
2		ovexd 6680	. . 3 |- ( ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) /\ ( x e. ( CC \ { 0 , 1 } ) /\ y e. ( CC \ { 0 } ) ) ) -> ( ( log ` y ) / ( log ` x ) ) e. _V )
3	2	ralrimivva 2971	. 2 |- ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> A. x e. ( CC \ { 0 , 1 } ) A. y e. ( CC \ { 0 } ) ( ( log ` y ) / ( log ` x ) ) e. _V )
4		cnex 10017	. . 3 |- CC e. _V
5		difexg 4808	. . 3 |- ( CC e. _V -> ( CC \ { 0 } ) e. _V )
6	4, 5	mp1i 13	. 2 |- ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> ( CC \ { 0 } ) e. _V )
7		eldifpr 4204	. . . 4 |- ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
8	7	biimpri 218	. . 3 |- ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) -> B e. ( CC \ { 0 , 1 } ) )
9	8	adantr 481	. 2 |- ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> B e. ( CC \ { 0 , 1 } ) )
10		simpr 477	. 2 |- ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> X e. ( CC \ { 0 } ) )
11	1, 3, 6, 9, 10	fvmpt2curryd 7397	1 |- ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> ( (curry logb ` B ) ` X ) = ( B logb X ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650  curry ccur 7391   CCcc 9934   0cc0 9936   1c1 9937   / cdiv 10684   logclog 24301   log_b clogb 24502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cur 7393  df-logb 24503

This theorem is referenced by:  relogbf  24529