Description: Closure of the general logarithm with a positive real base on positive reals. (Contributed by Stefan O'Rear, 19-Sep-2014) (Revised by Thierry Arnoux, 27-Sep-2017)
Ref | Expression | ||
---|---|---|---|
Assertion | relogbcl | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 | |
|
2 | 1 | rpcnne0d | |
3 | simp3 | |
|
4 | df-3an | |
|
5 | 2 3 4 | sylanbrc | |
6 | eldifpr | |
|
7 | 5 6 | sylibr | |
8 | simp2 | |
|
9 | 8 | rpcnne0d | |
10 | eldifsn | |
|
11 | 9 10 | sylibr | |
12 | logbval | |
|
13 | 7 11 12 | syl2anc | |
14 | relogcl | |
|
15 | 14 | 3ad2ant2 | |
16 | relogcl | |
|
17 | 16 | 3ad2ant1 | |
18 | logne0 | |
|
19 | 18 | 3adant2 | |
20 | 15 17 19 | redivcld | |
21 | 13 20 | eqeltrd | |