Description: Binomial coefficient: N choose 1 . (Contributed by NM, 21-Jun-2005) (Revised by Mario Carneiro, 8-Nov-2013)
Ref | Expression | ||
---|---|---|---|
Assertion | bcn1 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 | |
|
2 | 1eluzge0 | |
|
3 | 2 | a1i | |
4 | elnnuz | |
|
5 | 4 | biimpi | |
6 | elfzuzb | |
|
7 | 3 5 6 | sylanbrc | |
8 | bcval2 | |
|
9 | 7 8 | syl | |
10 | facnn2 | |
|
11 | fac1 | |
|
12 | 11 | oveq2i | |
13 | nnm1nn0 | |
|
14 | 13 | faccld | |
15 | 14 | nncnd | |
16 | 15 | mulid1d | |
17 | 12 16 | eqtrid | |
18 | 10 17 | oveq12d | |
19 | nncn | |
|
20 | 14 | nnne0d | |
21 | 19 15 20 | divcan3d | |
22 | 9 18 21 | 3eqtrd | |
23 | 0nn0 | |
|
24 | 1z | |
|
25 | 0lt1 | |
|
26 | 25 | olci | |
27 | bcval4 | |
|
28 | 23 24 26 27 | mp3an | |
29 | oveq1 | |
|
30 | eqeq12 | |
|
31 | 29 30 | mpancom | |
32 | 28 31 | mpbiri | |
33 | 22 32 | jaoi | |
34 | 1 33 | sylbi | |