Step |
Hyp |
Ref |
Expression |
1 |
|
fzfid |
|- ( N e. ( 0 ... A ) -> ( ( ( A - N ) + 1 ) ... A ) e. Fin ) |
2 |
|
elfzelz |
|- ( k e. ( ( ( A - N ) + 1 ) ... A ) -> k e. ZZ ) |
3 |
2
|
zcnd |
|- ( k e. ( ( ( A - N ) + 1 ) ... A ) -> k e. CC ) |
4 |
3
|
adantl |
|- ( ( N e. ( 0 ... A ) /\ k e. ( ( ( A - N ) + 1 ) ... A ) ) -> k e. CC ) |
5 |
1 4
|
fprodcl |
|- ( N e. ( 0 ... A ) -> prod_ k e. ( ( ( A - N ) + 1 ) ... A ) k e. CC ) |
6 |
|
fzfid |
|- ( N e. ( 0 ... A ) -> ( 1 ... ( A - N ) ) e. Fin ) |
7 |
|
elfznn |
|- ( k e. ( 1 ... ( A - N ) ) -> k e. NN ) |
8 |
7
|
adantl |
|- ( ( N e. ( 0 ... A ) /\ k e. ( 1 ... ( A - N ) ) ) -> k e. NN ) |
9 |
8
|
nncnd |
|- ( ( N e. ( 0 ... A ) /\ k e. ( 1 ... ( A - N ) ) ) -> k e. CC ) |
10 |
6 9
|
fprodcl |
|- ( N e. ( 0 ... A ) -> prod_ k e. ( 1 ... ( A - N ) ) k e. CC ) |
11 |
8
|
nnne0d |
|- ( ( N e. ( 0 ... A ) /\ k e. ( 1 ... ( A - N ) ) ) -> k =/= 0 ) |
12 |
6 9 11
|
fprodn0 |
|- ( N e. ( 0 ... A ) -> prod_ k e. ( 1 ... ( A - N ) ) k =/= 0 ) |
13 |
5 10 12
|
divcan3d |
|- ( N e. ( 0 ... A ) -> ( ( prod_ k e. ( 1 ... ( A - N ) ) k x. prod_ k e. ( ( ( A - N ) + 1 ) ... A ) k ) / prod_ k e. ( 1 ... ( A - N ) ) k ) = prod_ k e. ( ( ( A - N ) + 1 ) ... A ) k ) |
14 |
|
fznn0sub |
|- ( N e. ( 0 ... A ) -> ( A - N ) e. NN0 ) |
15 |
14
|
nn0red |
|- ( N e. ( 0 ... A ) -> ( A - N ) e. RR ) |
16 |
15
|
ltp1d |
|- ( N e. ( 0 ... A ) -> ( A - N ) < ( ( A - N ) + 1 ) ) |
17 |
|
fzdisj |
|- ( ( A - N ) < ( ( A - N ) + 1 ) -> ( ( 1 ... ( A - N ) ) i^i ( ( ( A - N ) + 1 ) ... A ) ) = (/) ) |
18 |
16 17
|
syl |
|- ( N e. ( 0 ... A ) -> ( ( 1 ... ( A - N ) ) i^i ( ( ( A - N ) + 1 ) ... A ) ) = (/) ) |
19 |
|
nn0p1nn |
|- ( ( A - N ) e. NN0 -> ( ( A - N ) + 1 ) e. NN ) |
20 |
14 19
|
syl |
|- ( N e. ( 0 ... A ) -> ( ( A - N ) + 1 ) e. NN ) |
21 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
22 |
20 21
|
eleqtrdi |
|- ( N e. ( 0 ... A ) -> ( ( A - N ) + 1 ) e. ( ZZ>= ` 1 ) ) |
23 |
14
|
nn0zd |
|- ( N e. ( 0 ... A ) -> ( A - N ) e. ZZ ) |
24 |
|
elfzel2 |
|- ( N e. ( 0 ... A ) -> A e. ZZ ) |
25 |
|
elfzle1 |
|- ( N e. ( 0 ... A ) -> 0 <_ N ) |
26 |
24
|
zred |
|- ( N e. ( 0 ... A ) -> A e. RR ) |
27 |
|
elfzelz |
|- ( N e. ( 0 ... A ) -> N e. ZZ ) |
28 |
27
|
zred |
|- ( N e. ( 0 ... A ) -> N e. RR ) |
29 |
26 28
|
subge02d |
|- ( N e. ( 0 ... A ) -> ( 0 <_ N <-> ( A - N ) <_ A ) ) |
30 |
25 29
|
mpbid |
|- ( N e. ( 0 ... A ) -> ( A - N ) <_ A ) |
31 |
|
eluz2 |
|- ( A e. ( ZZ>= ` ( A - N ) ) <-> ( ( A - N ) e. ZZ /\ A e. ZZ /\ ( A - N ) <_ A ) ) |
32 |
23 24 30 31
|
syl3anbrc |
|- ( N e. ( 0 ... A ) -> A e. ( ZZ>= ` ( A - N ) ) ) |
33 |
|
fzsplit2 |
|- ( ( ( ( A - N ) + 1 ) e. ( ZZ>= ` 1 ) /\ A e. ( ZZ>= ` ( A - N ) ) ) -> ( 1 ... A ) = ( ( 1 ... ( A - N ) ) u. ( ( ( A - N ) + 1 ) ... A ) ) ) |
34 |
22 32 33
|
syl2anc |
|- ( N e. ( 0 ... A ) -> ( 1 ... A ) = ( ( 1 ... ( A - N ) ) u. ( ( ( A - N ) + 1 ) ... A ) ) ) |
35 |
|
fzfid |
|- ( N e. ( 0 ... A ) -> ( 1 ... A ) e. Fin ) |
36 |
|
elfznn |
|- ( k e. ( 1 ... A ) -> k e. NN ) |
37 |
36
|
nncnd |
|- ( k e. ( 1 ... A ) -> k e. CC ) |
38 |
37
|
adantl |
|- ( ( N e. ( 0 ... A ) /\ k e. ( 1 ... A ) ) -> k e. CC ) |
39 |
18 34 35 38
|
fprodsplit |
|- ( N e. ( 0 ... A ) -> prod_ k e. ( 1 ... A ) k = ( prod_ k e. ( 1 ... ( A - N ) ) k x. prod_ k e. ( ( ( A - N ) + 1 ) ... A ) k ) ) |
40 |
39
|
oveq1d |
|- ( N e. ( 0 ... A ) -> ( prod_ k e. ( 1 ... A ) k / prod_ k e. ( 1 ... ( A - N ) ) k ) = ( ( prod_ k e. ( 1 ... ( A - N ) ) k x. prod_ k e. ( ( ( A - N ) + 1 ) ... A ) k ) / prod_ k e. ( 1 ... ( A - N ) ) k ) ) |
41 |
24
|
zcnd |
|- ( N e. ( 0 ... A ) -> A e. CC ) |
42 |
27
|
zcnd |
|- ( N e. ( 0 ... A ) -> N e. CC ) |
43 |
|
1cnd |
|- ( N e. ( 0 ... A ) -> 1 e. CC ) |
44 |
41 42 43
|
subsubd |
|- ( N e. ( 0 ... A ) -> ( A - ( N - 1 ) ) = ( ( A - N ) + 1 ) ) |
45 |
44
|
oveq1d |
|- ( N e. ( 0 ... A ) -> ( ( A - ( N - 1 ) ) ... A ) = ( ( ( A - N ) + 1 ) ... A ) ) |
46 |
45
|
prodeq1d |
|- ( N e. ( 0 ... A ) -> prod_ k e. ( ( A - ( N - 1 ) ) ... A ) k = prod_ k e. ( ( ( A - N ) + 1 ) ... A ) k ) |
47 |
13 40 46
|
3eqtr4rd |
|- ( N e. ( 0 ... A ) -> prod_ k e. ( ( A - ( N - 1 ) ) ... A ) k = ( prod_ k e. ( 1 ... A ) k / prod_ k e. ( 1 ... ( A - N ) ) k ) ) |
48 |
|
fallfacval3 |
|- ( N e. ( 0 ... A ) -> ( A FallFac N ) = prod_ k e. ( ( A - ( N - 1 ) ) ... A ) k ) |
49 |
|
elfz3nn0 |
|- ( N e. ( 0 ... A ) -> A e. NN0 ) |
50 |
|
fprodfac |
|- ( A e. NN0 -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k ) |
51 |
49 50
|
syl |
|- ( N e. ( 0 ... A ) -> ( ! ` A ) = prod_ k e. ( 1 ... A ) k ) |
52 |
|
fprodfac |
|- ( ( A - N ) e. NN0 -> ( ! ` ( A - N ) ) = prod_ k e. ( 1 ... ( A - N ) ) k ) |
53 |
14 52
|
syl |
|- ( N e. ( 0 ... A ) -> ( ! ` ( A - N ) ) = prod_ k e. ( 1 ... ( A - N ) ) k ) |
54 |
51 53
|
oveq12d |
|- ( N e. ( 0 ... A ) -> ( ( ! ` A ) / ( ! ` ( A - N ) ) ) = ( prod_ k e. ( 1 ... A ) k / prod_ k e. ( 1 ... ( A - N ) ) k ) ) |
55 |
47 48 54
|
3eqtr4d |
|- ( N e. ( 0 ... A ) -> ( A FallFac N ) = ( ( ! ` A ) / ( ! ` ( A - N ) ) ) ) |