Description: Separate out the first term in a finite product. (Contributed by Scott Fenton, 24-Dec-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | fprod1p.1 | |
|
fprod1p.2 | |
||
fprod1p.3 | |
||
Assertion | fprod1p | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprod1p.1 | |
|
2 | fprod1p.2 | |
|
3 | fprod1p.3 | |
|
4 | eluzfz1 | |
|
5 | 1 4 | syl | |
6 | 5 | elfzelzd | |
7 | fzsn | |
|
8 | 6 7 | syl | |
9 | 8 | ineq1d | |
10 | 6 | zred | |
11 | 10 | ltp1d | |
12 | fzdisj | |
|
13 | 11 12 | syl | |
14 | 9 13 | eqtr3d | |
15 | fzsplit | |
|
16 | 5 15 | syl | |
17 | 8 | uneq1d | |
18 | 16 17 | eqtrd | |
19 | fzfid | |
|
20 | 14 18 19 2 | fprodsplit | |
21 | 3 | eleq1d | |
22 | 2 | ralrimiva | |
23 | 21 22 5 | rspcdva | |
24 | 3 | prodsn | |
25 | 5 23 24 | syl2anc | |
26 | 25 | oveq1d | |
27 | 20 26 | eqtrd | |