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 ) ) |