Description: A submonoid of a submonoid is a submonoid. (Contributed by Mario Carneiro, 21-Jun-2015)
Ref | Expression | ||
---|---|---|---|
Hypothesis | subsubm.h | |
|
Assertion | subsubm | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subsubm.h | |
|
2 | eqid | |
|
3 | 2 | submss | |
4 | 3 | adantl | |
5 | 1 | submbas | |
6 | 5 | adantr | |
7 | 4 6 | sseqtrrd | |
8 | eqid | |
|
9 | 8 | submss | |
10 | 9 | adantr | |
11 | 7 10 | sstrd | |
12 | eqid | |
|
13 | 1 12 | subm0 | |
14 | 13 | adantr | |
15 | eqid | |
|
16 | 15 | subm0cl | |
17 | 16 | adantl | |
18 | 14 17 | eqeltrd | |
19 | 1 | oveq1i | |
20 | ressabs | |
|
21 | 19 20 | eqtrid | |
22 | 7 21 | syldan | |
23 | eqid | |
|
24 | 23 | submmnd | |
25 | 24 | adantl | |
26 | 22 25 | eqeltrrd | |
27 | submrcl | |
|
28 | 27 | adantr | |
29 | eqid | |
|
30 | 8 12 29 | issubm2 | |
31 | 28 30 | syl | |
32 | 11 18 26 31 | mpbir3and | |
33 | 32 7 | jca | |
34 | simprr | |
|
35 | 5 | adantr | |
36 | 34 35 | sseqtrd | |
37 | 13 | adantr | |
38 | 12 | subm0cl | |
39 | 38 | ad2antrl | |
40 | 37 39 | eqeltrrd | |
41 | 21 | adantrl | |
42 | 29 | submmnd | |
43 | 42 | ad2antrl | |
44 | 41 43 | eqeltrd | |
45 | 1 | submmnd | |
46 | 45 | adantr | |
47 | 2 15 23 | issubm2 | |
48 | 46 47 | syl | |
49 | 36 40 44 48 | mpbir3and | |
50 | 33 49 | impbida | |