Step |
Hyp |
Ref |
Expression |
1 |
|
peano2nn0 |
|- ( N e. NN0 -> ( N + 1 ) e. NN0 ) |
2 |
|
nn0z |
|- ( N e. NN0 -> N e. ZZ ) |
3 |
|
peano2zm |
|- ( N e. ZZ -> ( N - 1 ) e. ZZ ) |
4 |
2 3
|
syl |
|- ( N e. NN0 -> ( N - 1 ) e. ZZ ) |
5 |
|
bccmpl |
|- ( ( ( N + 1 ) e. NN0 /\ ( N - 1 ) e. ZZ ) -> ( ( N + 1 ) _C ( N - 1 ) ) = ( ( N + 1 ) _C ( ( N + 1 ) - ( N - 1 ) ) ) ) |
6 |
1 4 5
|
syl2anc |
|- ( N e. NN0 -> ( ( N + 1 ) _C ( N - 1 ) ) = ( ( N + 1 ) _C ( ( N + 1 ) - ( N - 1 ) ) ) ) |
7 |
|
nn0cn |
|- ( N e. NN0 -> N e. CC ) |
8 |
|
1cnd |
|- ( N e. NN0 -> 1 e. CC ) |
9 |
7 8 8
|
pnncand |
|- ( N e. NN0 -> ( ( N + 1 ) - ( N - 1 ) ) = ( 1 + 1 ) ) |
10 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
11 |
9 10
|
eqtr4di |
|- ( N e. NN0 -> ( ( N + 1 ) - ( N - 1 ) ) = 2 ) |
12 |
11
|
oveq2d |
|- ( N e. NN0 -> ( ( N + 1 ) _C ( ( N + 1 ) - ( N - 1 ) ) ) = ( ( N + 1 ) _C 2 ) ) |
13 |
|
bcn2 |
|- ( ( N + 1 ) e. NN0 -> ( ( N + 1 ) _C 2 ) = ( ( ( N + 1 ) x. ( ( N + 1 ) - 1 ) ) / 2 ) ) |
14 |
1 13
|
syl |
|- ( N e. NN0 -> ( ( N + 1 ) _C 2 ) = ( ( ( N + 1 ) x. ( ( N + 1 ) - 1 ) ) / 2 ) ) |
15 |
|
ax-1cn |
|- 1 e. CC |
16 |
|
pncan |
|- ( ( N e. CC /\ 1 e. CC ) -> ( ( N + 1 ) - 1 ) = N ) |
17 |
7 15 16
|
sylancl |
|- ( N e. NN0 -> ( ( N + 1 ) - 1 ) = N ) |
18 |
17
|
oveq2d |
|- ( N e. NN0 -> ( ( N + 1 ) x. ( ( N + 1 ) - 1 ) ) = ( ( N + 1 ) x. N ) ) |
19 |
18
|
oveq1d |
|- ( N e. NN0 -> ( ( ( N + 1 ) x. ( ( N + 1 ) - 1 ) ) / 2 ) = ( ( ( N + 1 ) x. N ) / 2 ) ) |
20 |
14 19
|
eqtrd |
|- ( N e. NN0 -> ( ( N + 1 ) _C 2 ) = ( ( ( N + 1 ) x. N ) / 2 ) ) |
21 |
12 20
|
eqtrd |
|- ( N e. NN0 -> ( ( N + 1 ) _C ( ( N + 1 ) - ( N - 1 ) ) ) = ( ( ( N + 1 ) x. N ) / 2 ) ) |
22 |
6 21
|
eqtrd |
|- ( N e. NN0 -> ( ( N + 1 ) _C ( N - 1 ) ) = ( ( ( N + 1 ) x. N ) / 2 ) ) |