Description: A finite product of integers is divisible by any of its factors. (Contributed by AV, 14-Aug-2020) (Proof shortened by AV, 2-Aug-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | fproddvdsd.f | |
|
fproddvdsd.s | |
||
Assertion | fproddvdsd | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fproddvdsd.f | |
|
2 | fproddvdsd.s | |
|
3 | f1oi | |
|
4 | f1of | |
|
5 | 3 4 | mp1i | |
6 | 1 2 5 | fprodfvdvdsd | |
7 | 2 | sselda | |
8 | fvresi | |
|
9 | 7 8 | syl | |
10 | 9 | eqcomd | |
11 | 2 | sseld | |
12 | 11 | adantr | |
13 | 12 | imp | |
14 | fvresi | |
|
15 | 13 14 | syl | |
16 | 15 | eqcomd | |
17 | 16 | prodeq2dv | |
18 | 10 17 | breq12d | |
19 | 18 | ralbidva | |
20 | 6 19 | mpbird | |