Step |
Hyp |
Ref |
Expression |
1 |
|
elfz3nn0 |
|- ( K e. ( 0 ... N ) -> N e. NN0 ) |
2 |
|
facp1 |
|- ( N e. NN0 -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) ) |
3 |
1 2
|
syl |
|- ( K e. ( 0 ... N ) -> ( ! ` ( N + 1 ) ) = ( ( ! ` N ) x. ( N + 1 ) ) ) |
4 |
|
fznn0sub |
|- ( K e. ( 0 ... N ) -> ( N - K ) e. NN0 ) |
5 |
|
facp1 |
|- ( ( N - K ) e. NN0 -> ( ! ` ( ( N - K ) + 1 ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) ) |
6 |
4 5
|
syl |
|- ( K e. ( 0 ... N ) -> ( ! ` ( ( N - K ) + 1 ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) ) |
7 |
1
|
nn0cnd |
|- ( K e. ( 0 ... N ) -> N e. CC ) |
8 |
|
1cnd |
|- ( K e. ( 0 ... N ) -> 1 e. CC ) |
9 |
|
elfznn0 |
|- ( K e. ( 0 ... N ) -> K e. NN0 ) |
10 |
9
|
nn0cnd |
|- ( K e. ( 0 ... N ) -> K e. CC ) |
11 |
7 8 10
|
addsubd |
|- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) = ( ( N - K ) + 1 ) ) |
12 |
11
|
fveq2d |
|- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ! ` ( ( N - K ) + 1 ) ) ) |
13 |
11
|
oveq2d |
|- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N - K ) + 1 ) ) ) |
14 |
6 12 13
|
3eqtr4d |
|- ( K e. ( 0 ... N ) -> ( ! ` ( ( N + 1 ) - K ) ) = ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) ) |
15 |
14
|
oveq1d |
|- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) |
16 |
4
|
faccld |
|- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. NN ) |
17 |
16
|
nncnd |
|- ( K e. ( 0 ... N ) -> ( ! ` ( N - K ) ) e. CC ) |
18 |
|
nn0p1nn |
|- ( ( N - K ) e. NN0 -> ( ( N - K ) + 1 ) e. NN ) |
19 |
4 18
|
syl |
|- ( K e. ( 0 ... N ) -> ( ( N - K ) + 1 ) e. NN ) |
20 |
11 19
|
eqeltrd |
|- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. NN ) |
21 |
20
|
nncnd |
|- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) e. CC ) |
22 |
9
|
faccld |
|- ( K e. ( 0 ... N ) -> ( ! ` K ) e. NN ) |
23 |
22
|
nncnd |
|- ( K e. ( 0 ... N ) -> ( ! ` K ) e. CC ) |
24 |
17 21 23
|
mul32d |
|- ( K e. ( 0 ... N ) -> ( ( ( ! ` ( N - K ) ) x. ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) ) |
25 |
15 24
|
eqtrd |
|- ( K e. ( 0 ... N ) -> ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) = ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) ) |
26 |
3 25
|
oveq12d |
|- ( K e. ( 0 ... N ) -> ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) = ( ( ( ! ` N ) x. ( N + 1 ) ) / ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) ) ) |
27 |
1
|
faccld |
|- ( K e. ( 0 ... N ) -> ( ! ` N ) e. NN ) |
28 |
27
|
nncnd |
|- ( K e. ( 0 ... N ) -> ( ! ` N ) e. CC ) |
29 |
|
nn0p1nn |
|- ( N e. NN0 -> ( N + 1 ) e. NN ) |
30 |
1 29
|
syl |
|- ( K e. ( 0 ... N ) -> ( N + 1 ) e. NN ) |
31 |
30
|
nncnd |
|- ( K e. ( 0 ... N ) -> ( N + 1 ) e. CC ) |
32 |
16 22
|
nnmulcld |
|- ( K e. ( 0 ... N ) -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN ) |
33 |
|
nncn |
|- ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC ) |
34 |
|
nnne0 |
|- ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN -> ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) =/= 0 ) |
35 |
33 34
|
jca |
|- ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. NN -> ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC /\ ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) =/= 0 ) ) |
36 |
32 35
|
syl |
|- ( K e. ( 0 ... N ) -> ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC /\ ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) =/= 0 ) ) |
37 |
20
|
nnne0d |
|- ( K e. ( 0 ... N ) -> ( ( N + 1 ) - K ) =/= 0 ) |
38 |
21 37
|
jca |
|- ( K e. ( 0 ... N ) -> ( ( ( N + 1 ) - K ) e. CC /\ ( ( N + 1 ) - K ) =/= 0 ) ) |
39 |
|
divmuldiv |
|- ( ( ( ( ! ` N ) e. CC /\ ( N + 1 ) e. CC ) /\ ( ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) e. CC /\ ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) =/= 0 ) /\ ( ( ( N + 1 ) - K ) e. CC /\ ( ( N + 1 ) - K ) =/= 0 ) ) ) -> ( ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) = ( ( ( ! ` N ) x. ( N + 1 ) ) / ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) ) ) |
40 |
28 31 36 38 39
|
syl22anc |
|- ( K e. ( 0 ... N ) -> ( ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) = ( ( ( ! ` N ) x. ( N + 1 ) ) / ( ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) x. ( ( N + 1 ) - K ) ) ) ) |
41 |
26 40
|
eqtr4d |
|- ( K e. ( 0 ... N ) -> ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) = ( ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) ) |
42 |
|
fzelp1 |
|- ( K e. ( 0 ... N ) -> K e. ( 0 ... ( N + 1 ) ) ) |
43 |
|
bcval2 |
|- ( K e. ( 0 ... ( N + 1 ) ) -> ( ( N + 1 ) _C K ) = ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) ) |
44 |
42 43
|
syl |
|- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( ! ` ( N + 1 ) ) / ( ( ! ` ( ( N + 1 ) - K ) ) x. ( ! ` K ) ) ) ) |
45 |
|
bcval2 |
|- ( K e. ( 0 ... N ) -> ( N _C K ) = ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) ) |
46 |
45
|
oveq1d |
|- ( K e. ( 0 ... N ) -> ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) = ( ( ( ! ` N ) / ( ( ! ` ( N - K ) ) x. ( ! ` K ) ) ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) ) |
47 |
41 44 46
|
3eqtr4d |
|- ( K e. ( 0 ... N ) -> ( ( N + 1 ) _C K ) = ( ( N _C K ) x. ( ( N + 1 ) / ( ( N + 1 ) - K ) ) ) ) |