Description: Compute the binomial coefficient " N choose 2 " from " ( N - 1 ) choose 2 ": (N-1) + ( (N-1) 2 ) = ( N 2 ). (Contributed by Alexander van der Vekens, 7-Jan-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | bcn2m1 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnm1nn0 | |
|
2 | 1 | nn0cnd | |
3 | 2z | |
|
4 | bccl | |
|
5 | 1 3 4 | sylancl | |
6 | 5 | nn0cnd | |
7 | 2 6 | addcomd | |
8 | bcn1 | |
|
9 | 8 | eqcomd | |
10 | 1 9 | syl | |
11 | 1e2m1 | |
|
12 | 11 | a1i | |
13 | 12 | oveq2d | |
14 | 10 13 | eqtrd | |
15 | 14 | oveq2d | |
16 | bcpasc | |
|
17 | 1 3 16 | sylancl | |
18 | nncn | |
|
19 | 1cnd | |
|
20 | 18 19 | npcand | |
21 | 20 | oveq1d | |
22 | 17 21 | eqtrd | |
23 | 7 15 22 | 3eqtrd | |