Description: A finite product of integers is divisible by any of its factors being function values. (Contributed by AV, 1-Aug-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | fprodfvdvdsd.a | |
|
fprodfvdvdsd.b | |
||
fprodfvdvdsd.f | |
||
Assertion | fprodfvdvdsd | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprodfvdvdsd.a | |
|
2 | fprodfvdvdsd.b | |
|
3 | fprodfvdvdsd.f | |
|
4 | 1 | adantr | |
5 | diffi | |
|
6 | 4 5 | syl | |
7 | 3 | adantr | |
8 | 2 | ssdifssd | |
9 | 8 | sselda | |
10 | 7 9 | ffvelrnd | |
11 | 10 | adantlr | |
12 | 6 11 | fprodzcl | |
13 | 3 | adantr | |
14 | 2 | sselda | |
15 | 13 14 | ffvelrnd | |
16 | dvdsmul2 | |
|
17 | 12 15 16 | syl2anc | |
18 | 17 | ralrimiva | |
19 | neldifsnd | |
|
20 | disjsn | |
|
21 | 19 20 | sylibr | |
22 | difsnid | |
|
23 | 22 | eqcomd | |
24 | 23 | adantl | |
25 | 13 | adantr | |
26 | 2 | adantr | |
27 | 26 | sselda | |
28 | 25 27 | ffvelrnd | |
29 | 28 | zcnd | |
30 | 21 24 4 29 | fprodsplit | |
31 | simpr | |
|
32 | 15 | zcnd | |
33 | fveq2 | |
|
34 | 33 | prodsn | |
35 | 31 32 34 | syl2anc | |
36 | 35 | oveq2d | |
37 | 30 36 | eqtrd | |
38 | 37 | breq2d | |
39 | 38 | ralbidva | |
40 | 18 39 | mpbird | |