Step |
Hyp |
Ref |
Expression |
1 |
|
peano2nn0 |
|- ( N e. NN0 -> ( N + 1 ) e. NN0 ) |
2 |
|
nn0z |
|- ( N e. NN0 -> N e. ZZ ) |
3 |
|
bccmpl |
|- ( ( ( N + 1 ) e. NN0 /\ N e. ZZ ) -> ( ( N + 1 ) _C N ) = ( ( N + 1 ) _C ( ( N + 1 ) - N ) ) ) |
4 |
1 2 3
|
syl2anc |
|- ( N e. NN0 -> ( ( N + 1 ) _C N ) = ( ( N + 1 ) _C ( ( N + 1 ) - N ) ) ) |
5 |
|
nn0cn |
|- ( N e. NN0 -> N e. CC ) |
6 |
|
ax-1cn |
|- 1 e. CC |
7 |
|
pncan2 |
|- ( ( N e. CC /\ 1 e. CC ) -> ( ( N + 1 ) - N ) = 1 ) |
8 |
5 6 7
|
sylancl |
|- ( N e. NN0 -> ( ( N + 1 ) - N ) = 1 ) |
9 |
8
|
oveq2d |
|- ( N e. NN0 -> ( ( N + 1 ) _C ( ( N + 1 ) - N ) ) = ( ( N + 1 ) _C 1 ) ) |
10 |
|
bcn1 |
|- ( ( N + 1 ) e. NN0 -> ( ( N + 1 ) _C 1 ) = ( N + 1 ) ) |
11 |
1 10
|
syl |
|- ( N e. NN0 -> ( ( N + 1 ) _C 1 ) = ( N + 1 ) ) |
12 |
4 9 11
|
3eqtrd |
|- ( N e. NN0 -> ( ( N + 1 ) _C N ) = ( N + 1 ) ) |