Description: The closed subspaces of a topology-bearing module form a complete lattice. Demonstration for mreclatBAD . (Contributed by Stefan O'Rear, 31-Jan-2015) TODO ( df-riota update): This proof uses the old df-clat and references the required instance of mreclatBAD as a hypothesis. When mreclatBAD is corrected to become mreclat, delete this theorem and uncomment the mreclatdemo below.
Ref | Expression | ||
---|---|---|---|
Hypothesis | mreclatBAD. | |
|
Assertion | mreclatdemoBAD | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mreclatBAD. | |
|
2 | fvex | |
|
3 | 2 | uniex | |
4 | mremre | |
|
5 | 3 4 | mp1i | |
6 | elinel2 | |
|
7 | eqid | |
|
8 | eqid | |
|
9 | 7 8 | lssmre | |
10 | 6 9 | syl | |
11 | elinel1 | |
|
12 | eqid | |
|
13 | 7 12 | tpsuni | |
14 | 13 | fveq2d | |
15 | 11 14 | syl | |
16 | 10 15 | eleqtrd | |
17 | 12 | tpstop | |
18 | eqid | |
|
19 | 18 | cldmre | |
20 | 11 17 19 | 3syl | |
21 | mreincl | |
|
22 | 5 16 20 21 | syl3anc | |
23 | 22 1 | syl | |