Description: A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010)
Ref | Expression | ||
---|---|---|---|
Hypotheses | abl4pnp.1 | |
|
abl4pnp.2 | |
||
Assertion | ablo4pnp | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abl4pnp.1 | |
|
2 | abl4pnp.2 | |
|
3 | df-3an | |
|
4 | 1 2 | ablomuldiv | |
5 | 3 4 | sylan2br | |
6 | 5 | adantrrr | |
7 | 6 | oveq1d | |
8 | ablogrpo | |
|
9 | 1 | grpocl | |
10 | 9 | 3expib | |
11 | 8 10 | syl | |
12 | 11 | anim1d | |
13 | 3anass | |
|
14 | 12 13 | syl6ibr | |
15 | 14 | imp | |
16 | 1 2 | ablodivdiv4 | |
17 | 15 16 | syldan | |
18 | 1 2 | grpodivcl | |
19 | 18 | 3expib | |
20 | 19 | anim1d | |
21 | an4 | |
|
22 | 3anass | |
|
23 | 20 21 22 | 3imtr4g | |
24 | 23 | imp | |
25 | 1 2 | grpomuldivass | |
26 | 24 25 | syldan | |
27 | 8 26 | sylan | |
28 | 7 17 27 | 3eqtr3d | |