Description: Closure of addition of integers. (Contributed by NM, 9-May-2004) (Proof shortened by Mario Carneiro, 16-May-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | zaddcl | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz2 | |
|
2 | elz2 | |
|
3 | reeanv | |
|
4 | reeanv | |
|
5 | nnaddcl | |
|
6 | 5 | adantr | |
7 | nnaddcl | |
|
8 | 7 | adantl | |
9 | nncn | |
|
10 | nncn | |
|
11 | 9 10 | anim12i | |
12 | nncn | |
|
13 | nncn | |
|
14 | 12 13 | anim12i | |
15 | addsub4 | |
|
16 | 11 14 15 | syl2an | |
17 | 16 | eqcomd | |
18 | rspceov | |
|
19 | 6 8 17 18 | syl3anc | |
20 | elz2 | |
|
21 | 19 20 | sylibr | |
22 | oveq12 | |
|
23 | 22 | eleq1d | |
24 | 21 23 | syl5ibrcom | |
25 | 24 | rexlimdvva | |
26 | 4 25 | syl5bir | |
27 | 26 | rexlimivv | |
28 | 3 27 | sylbir | |
29 | 1 2 28 | syl2anb | |