Description: Distribution of multiplication over subtraction. Theorem I.5 of Apostol p. 18. (Contributed by NM, 18-Nov-2004)
Ref | Expression | ||
---|---|---|---|
Assertion | subdi | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 | |
|
2 | simp3 | |
|
3 | subcl | |
|
4 | 3 | 3adant1 | |
5 | 1 2 4 | adddid | |
6 | pncan3 | |
|
7 | 6 | ancoms | |
8 | 7 | 3adant1 | |
9 | 8 | oveq2d | |
10 | 5 9 | eqtr3d | |
11 | mulcl | |
|
12 | 11 | 3adant3 | |
13 | mulcl | |
|
14 | 13 | 3adant2 | |
15 | mulcl | |
|
16 | 3 15 | sylan2 | |
17 | 16 | 3impb | |
18 | 12 14 17 | subaddd | |
19 | 10 18 | mpbird | |
20 | 19 | eqcomd | |