Description: A lower bounded real function is eventually bounded iff it is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | o1lo1.1 | |
|
o1lo12.2 | |
||
o1lo12.3 | |
||
Assertion | o1lo12 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | o1lo1.1 | |
|
2 | o1lo12.2 | |
|
3 | o1lo12.3 | |
|
4 | o1dm | |
|
5 | 4 | a1i | |
6 | lo1dm | |
|
7 | 6 | a1i | |
8 | 1 | ralrimiva | |
9 | dmmptg | |
|
10 | 8 9 | syl | |
11 | 10 | sseq1d | |
12 | simpr | |
|
13 | 1 | renegcld | |
14 | 13 | adantlr | |
15 | 2 | adantr | |
16 | 15 | renegcld | |
17 | 2 | adantr | |
18 | 17 1 | lenegd | |
19 | 3 18 | mpbid | |
20 | 19 | ad2ant2r | |
21 | 12 14 15 16 20 | ello1d | |
22 | 1 | o1lo1 | |
23 | 22 | rbaibd | |
24 | 21 23 | syldan | |
25 | 24 | ex | |
26 | 11 25 | sylbid | |
27 | 5 7 26 | pm5.21ndd | |