Step |
Hyp |
Ref |
Expression |
1 |
|
nnm1nn0 |
|- ( N e. NN -> ( N - 1 ) e. NN0 ) |
2 |
1
|
nn0cnd |
|- ( N e. NN -> ( N - 1 ) e. CC ) |
3 |
|
2z |
|- 2 e. ZZ |
4 |
|
bccl |
|- ( ( ( N - 1 ) e. NN0 /\ 2 e. ZZ ) -> ( ( N - 1 ) _C 2 ) e. NN0 ) |
5 |
1 3 4
|
sylancl |
|- ( N e. NN -> ( ( N - 1 ) _C 2 ) e. NN0 ) |
6 |
5
|
nn0cnd |
|- ( N e. NN -> ( ( N - 1 ) _C 2 ) e. CC ) |
7 |
2 6
|
addcomd |
|- ( N e. NN -> ( ( N - 1 ) + ( ( N - 1 ) _C 2 ) ) = ( ( ( N - 1 ) _C 2 ) + ( N - 1 ) ) ) |
8 |
|
bcn1 |
|- ( ( N - 1 ) e. NN0 -> ( ( N - 1 ) _C 1 ) = ( N - 1 ) ) |
9 |
8
|
eqcomd |
|- ( ( N - 1 ) e. NN0 -> ( N - 1 ) = ( ( N - 1 ) _C 1 ) ) |
10 |
1 9
|
syl |
|- ( N e. NN -> ( N - 1 ) = ( ( N - 1 ) _C 1 ) ) |
11 |
|
1e2m1 |
|- 1 = ( 2 - 1 ) |
12 |
11
|
a1i |
|- ( N e. NN -> 1 = ( 2 - 1 ) ) |
13 |
12
|
oveq2d |
|- ( N e. NN -> ( ( N - 1 ) _C 1 ) = ( ( N - 1 ) _C ( 2 - 1 ) ) ) |
14 |
10 13
|
eqtrd |
|- ( N e. NN -> ( N - 1 ) = ( ( N - 1 ) _C ( 2 - 1 ) ) ) |
15 |
14
|
oveq2d |
|- ( N e. NN -> ( ( ( N - 1 ) _C 2 ) + ( N - 1 ) ) = ( ( ( N - 1 ) _C 2 ) + ( ( N - 1 ) _C ( 2 - 1 ) ) ) ) |
16 |
|
bcpasc |
|- ( ( ( N - 1 ) e. NN0 /\ 2 e. ZZ ) -> ( ( ( N - 1 ) _C 2 ) + ( ( N - 1 ) _C ( 2 - 1 ) ) ) = ( ( ( N - 1 ) + 1 ) _C 2 ) ) |
17 |
1 3 16
|
sylancl |
|- ( N e. NN -> ( ( ( N - 1 ) _C 2 ) + ( ( N - 1 ) _C ( 2 - 1 ) ) ) = ( ( ( N - 1 ) + 1 ) _C 2 ) ) |
18 |
|
nncn |
|- ( N e. NN -> N e. CC ) |
19 |
|
1cnd |
|- ( N e. NN -> 1 e. CC ) |
20 |
18 19
|
npcand |
|- ( N e. NN -> ( ( N - 1 ) + 1 ) = N ) |
21 |
20
|
oveq1d |
|- ( N e. NN -> ( ( ( N - 1 ) + 1 ) _C 2 ) = ( N _C 2 ) ) |
22 |
17 21
|
eqtrd |
|- ( N e. NN -> ( ( ( N - 1 ) _C 2 ) + ( ( N - 1 ) _C ( 2 - 1 ) ) ) = ( N _C 2 ) ) |
23 |
7 15 22
|
3eqtrd |
|- ( N e. NN -> ( ( N - 1 ) + ( ( N - 1 ) _C 2 ) ) = ( N _C 2 ) ) |