Step |
Hyp |
Ref |
Expression |
1 |
|
zgt1rpn0n1 |
|- ( B e. ( ZZ>= ` 2 ) -> ( B e. RR+ /\ B =/= 0 /\ B =/= 1 ) ) |
2 |
1
|
adantr |
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B e. RR+ /\ B =/= 0 /\ B =/= 1 ) ) |
3 |
2
|
simp1d |
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> B e. RR+ ) |
4 |
3
|
rpcnd |
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> B e. CC ) |
5 |
2
|
simp2d |
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> B =/= 0 ) |
6 |
2
|
simp3d |
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> B =/= 1 ) |
7 |
|
eldifpr |
|- ( B e. ( CC \ { 0 , 1 } ) <-> ( B e. CC /\ B =/= 0 /\ B =/= 1 ) ) |
8 |
4 5 6 7
|
syl3anbrc |
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> B e. ( CC \ { 0 , 1 } ) ) |
9 |
|
simpr |
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> X e. RR+ ) |
10 |
9
|
rpcnne0d |
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( X e. CC /\ X =/= 0 ) ) |
11 |
|
eldifsn |
|- ( X e. ( CC \ { 0 } ) <-> ( X e. CC /\ X =/= 0 ) ) |
12 |
10 11
|
sylibr |
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> X e. ( CC \ { 0 } ) ) |
13 |
|
logbval |
|- ( ( B e. ( CC \ { 0 , 1 } ) /\ X e. ( CC \ { 0 } ) ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) ) |
14 |
8 12 13
|
syl2anc |
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) ) |