Step |
Hyp |
Ref |
Expression |
1 |
|
fzctr |
|- ( N e. NN0 -> N e. ( 0 ... ( 2 x. N ) ) ) |
2 |
|
bcval2 |
|- ( N e. ( 0 ... ( 2 x. N ) ) -> ( ( 2 x. N ) _C N ) = ( ( ! ` ( 2 x. N ) ) / ( ( ! ` ( ( 2 x. N ) - N ) ) x. ( ! ` N ) ) ) ) |
3 |
1 2
|
syl |
|- ( N e. NN0 -> ( ( 2 x. N ) _C N ) = ( ( ! ` ( 2 x. N ) ) / ( ( ! ` ( ( 2 x. N ) - N ) ) x. ( ! ` N ) ) ) ) |
4 |
|
nn0cn |
|- ( N e. NN0 -> N e. CC ) |
5 |
4
|
2timesd |
|- ( N e. NN0 -> ( 2 x. N ) = ( N + N ) ) |
6 |
4 4 5
|
mvrladdd |
|- ( N e. NN0 -> ( ( 2 x. N ) - N ) = N ) |
7 |
6
|
fveq2d |
|- ( N e. NN0 -> ( ! ` ( ( 2 x. N ) - N ) ) = ( ! ` N ) ) |
8 |
7
|
oveq1d |
|- ( N e. NN0 -> ( ( ! ` ( ( 2 x. N ) - N ) ) x. ( ! ` N ) ) = ( ( ! ` N ) x. ( ! ` N ) ) ) |
9 |
8
|
oveq2d |
|- ( N e. NN0 -> ( ( ! ` ( 2 x. N ) ) / ( ( ! ` ( ( 2 x. N ) - N ) ) x. ( ! ` N ) ) ) = ( ( ! ` ( 2 x. N ) ) / ( ( ! ` N ) x. ( ! ` N ) ) ) ) |
10 |
3 9
|
eqtrd |
|- ( N e. NN0 -> ( ( 2 x. N ) _C N ) = ( ( ! ` ( 2 x. N ) ) / ( ( ! ` N ) x. ( ! ` N ) ) ) ) |