Description: The sequence of partial products of a finite product converges to the whole product. (Contributed by Scott Fenton, 4-Dec-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | prodmo.1 | |
|
prodmo.2 | |
||
prodrb.3 | |
||
fprodcvg.4 | |
||
Assertion | fprodcvg | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prodmo.1 | |
|
2 | prodmo.2 | |
|
3 | prodrb.3 | |
|
4 | fprodcvg.4 | |
|
5 | eqid | |
|
6 | eluzelz | |
|
7 | 3 6 | syl | |
8 | seqex | |
|
9 | 8 | a1i | |
10 | eqid | |
|
11 | eluzel2 | |
|
12 | 3 11 | syl | |
13 | eluzelz | |
|
14 | 13 | adantl | |
15 | iftrue | |
|
16 | 15 | adantl | |
17 | 2 | adantlr | |
18 | 16 17 | eqeltrd | |
19 | 18 | ex | |
20 | iffalse | |
|
21 | ax-1cn | |
|
22 | 20 21 | eqeltrdi | |
23 | 19 22 | pm2.61d1 | |
24 | 1 | fvmpt2 | |
25 | 14 23 24 | syl2anc | |
26 | 25 23 | eqeltrd | |
27 | 10 12 26 | prodf | |
28 | 27 3 | ffvelrnd | |
29 | mulid1 | |
|
30 | 29 | adantl | |
31 | 3 | adantr | |
32 | simpr | |
|
33 | 12 | adantr | |
34 | 26 | adantlr | |
35 | 10 33 34 | prodf | |
36 | 35 31 | ffvelrnd | |
37 | elfzuz | |
|
38 | eluzelz | |
|
39 | 38 | adantl | |
40 | 4 | sseld | |
41 | fznuz | |
|
42 | 40 41 | syl6 | |
43 | 42 | con2d | |
44 | 43 | imp | |
45 | 39 44 | eldifd | |
46 | fveqeq2 | |
|
47 | eldifi | |
|
48 | eldifn | |
|
49 | 48 20 | syl | |
50 | 49 21 | eqeltrdi | |
51 | 47 50 24 | syl2anc | |
52 | 51 49 | eqtrd | |
53 | 46 52 | vtoclga | |
54 | 45 53 | syl | |
55 | 37 54 | sylan2 | |
56 | 55 | adantlr | |
57 | 30 31 32 36 56 | seqid2 | |
58 | 57 | eqcomd | |
59 | 5 7 9 28 58 | climconst | |