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Metamath Proof Explorer

Theorem logbfval

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

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

Proof

Step Hyp Ref Expression
1 df-logb 
 |-  logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
2 ovexd 
 |-  ( ( ( ( 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 
 |-  ( ( ( 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 
 |-  CC e. _V
5 difexg 
 |-  ( CC e. _V -> ( CC \ { 0 } ) e. _V )
6 4 5 mp1i 
 |-  ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> ( CC \ { 0 } ) e. _V )
7 eldifpr 
 |-  ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) )
8 7 biimpri 
 |-  ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) -> B e. ( CC \ { 0 , 1 } ) )
9 8 adantr 
 |-  ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> B e. ( CC \ { 0 , 1 } ) )
10 simpr 
 |-  ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> X e. ( CC \ { 0 } ) )
11 1 3 6 9 10 fvmpocurryd 
 |-  ( ( ( B e. CC /\ B =/= 0 /\ B =/= 1 ) /\ X e. ( CC \ { 0 } ) ) -> ( ( curry logb ` B ) ` X ) = ( B logb X ) )