Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> A e. RR+ ) |
2 |
1
|
rpreccld |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( 1 / A ) e. RR+ ) |
3 |
|
relogbval |
|- ( ( B e. ( ZZ>= ` 2 ) /\ ( 1 / A ) e. RR+ ) -> ( B logb ( 1 / A ) ) = ( ( log ` ( 1 / A ) ) / ( log ` B ) ) ) |
4 |
2 3
|
syldan |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( B logb ( 1 / A ) ) = ( ( log ` ( 1 / A ) ) / ( log ` B ) ) ) |
5 |
|
relogbval |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( B logb A ) = ( ( log ` A ) / ( log ` B ) ) ) |
6 |
5
|
negeqd |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> -u ( B logb A ) = -u ( ( log ` A ) / ( log ` B ) ) ) |
7 |
1
|
rpcnd |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> A e. CC ) |
8 |
1
|
rpne0d |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> A =/= 0 ) |
9 |
7 8
|
logcld |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` A ) e. CC ) |
10 |
|
zgt1rpn0n1 |
|- ( B e. ( ZZ>= ` 2 ) -> ( B e. RR+ /\ B =/= 0 /\ B =/= 1 ) ) |
11 |
10
|
simp1d |
|- ( B e. ( ZZ>= ` 2 ) -> B e. RR+ ) |
12 |
11
|
adantr |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> B e. RR+ ) |
13 |
12
|
relogcld |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` B ) e. RR ) |
14 |
13
|
recnd |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` B ) e. CC ) |
15 |
10
|
simp3d |
|- ( B e. ( ZZ>= ` 2 ) -> B =/= 1 ) |
16 |
15
|
adantr |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> B =/= 1 ) |
17 |
|
logne0 |
|- ( ( B e. RR+ /\ B =/= 1 ) -> ( log ` B ) =/= 0 ) |
18 |
12 16 17
|
syl2anc |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` B ) =/= 0 ) |
19 |
9 14 18
|
divnegd |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> -u ( ( log ` A ) / ( log ` B ) ) = ( -u ( log ` A ) / ( log ` B ) ) ) |
20 |
7 8
|
reccld |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( 1 / A ) e. CC ) |
21 |
7 8
|
recne0d |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( 1 / A ) =/= 0 ) |
22 |
20 21
|
logcld |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` ( 1 / A ) ) e. CC ) |
23 |
1
|
relogcld |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` A ) e. RR ) |
24 |
23
|
reim0d |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( Im ` ( log ` A ) ) = 0 ) |
25 |
|
0re |
|- 0 e. RR |
26 |
|
pipos |
|- 0 < _pi |
27 |
25 26
|
gtneii |
|- _pi =/= 0 |
28 |
27
|
a1i |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> _pi =/= 0 ) |
29 |
28
|
necomd |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> 0 =/= _pi ) |
30 |
24 29
|
eqnetrd |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( Im ` ( log ` A ) ) =/= _pi ) |
31 |
|
logrec |
|- ( ( A e. CC /\ A =/= 0 /\ ( Im ` ( log ` A ) ) =/= _pi ) -> ( log ` A ) = -u ( log ` ( 1 / A ) ) ) |
32 |
7 8 30 31
|
syl3anc |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( log ` A ) = -u ( log ` ( 1 / A ) ) ) |
33 |
32
|
eqcomd |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> -u ( log ` ( 1 / A ) ) = ( log ` A ) ) |
34 |
22 33
|
negcon1ad |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> -u ( log ` A ) = ( log ` ( 1 / A ) ) ) |
35 |
34
|
oveq1d |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( -u ( log ` A ) / ( log ` B ) ) = ( ( log ` ( 1 / A ) ) / ( log ` B ) ) ) |
36 |
6 19 35
|
3eqtrd |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> -u ( B logb A ) = ( ( log ` ( 1 / A ) ) / ( log ` B ) ) ) |
37 |
4 36
|
eqtr4d |
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( B logb ( 1 / A ) ) = -u ( B logb A ) ) |