Description: The least upper bound law for Hilbert subspace sum. (Contributed by NM, 15-Jun-2004) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | shslub.1 | |
|
shslub.2 | |
||
shslub.3 | |
||
Assertion | shslubi | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | shslub.1 | |
|
2 | shslub.2 | |
|
3 | shslub.3 | |
|
4 | 1 3 2 | shlessi | |
5 | 3 2 | shscomi | |
6 | 4 5 | sseqtrdi | |
7 | 2 3 3 | shlessi | |
8 | 3 | shsidmi | |
9 | 7 8 | sseqtrdi | |
10 | 6 9 | sylan9ss | |
11 | 1 2 | shsub1i | |
12 | sstr | |
|
13 | 11 12 | mpan | |
14 | 2 1 | shsub2i | |
15 | sstr | |
|
16 | 14 15 | mpan | |
17 | 13 16 | jca | |
18 | 10 17 | impbii | |