Description: An associative/commutative law for division and multiplication. (Contributed by AV, 10-Jul-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | divmulasscom | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divmulass | |
|
2 | mulcom | |
|
3 | 2 | 3adant3 | |
4 | 3 | adantr | |
5 | 4 | oveq1d | |
6 | simpl2 | |
|
7 | simpl1 | |
|
8 | simp3 | |
|
9 | 8 | anim1i | |
10 | 3anass | |
|
11 | 9 10 | sylibr | |
12 | divcl | |
|
13 | 11 12 | syl | |
14 | 6 7 13 | mulassd | |
15 | 8 | adantr | |
16 | simpr | |
|
17 | divass | |
|
18 | 7 15 16 17 | syl3anc | |
19 | 18 | eqcomd | |
20 | 19 | oveq2d | |
21 | 14 20 | eqtrd | |
22 | 1 5 21 | 3eqtrd | |