Step |
Hyp |
Ref |
Expression |
1 |
|
oddz |
|- ( N e. Odd -> N e. ZZ ) |
2 |
1
|
zcnd |
|- ( N e. Odd -> N e. CC ) |
3 |
|
npcan1 |
|- ( N e. CC -> ( ( N - 1 ) + 1 ) = N ) |
4 |
3
|
eqcomd |
|- ( N e. CC -> N = ( ( N - 1 ) + 1 ) ) |
5 |
4
|
oveq1d |
|- ( N e. CC -> ( N / 2 ) = ( ( ( N - 1 ) + 1 ) / 2 ) ) |
6 |
|
peano2cnm |
|- ( N e. CC -> ( N - 1 ) e. CC ) |
7 |
|
1cnd |
|- ( N e. CC -> 1 e. CC ) |
8 |
|
2cnne0 |
|- ( 2 e. CC /\ 2 =/= 0 ) |
9 |
8
|
a1i |
|- ( N e. CC -> ( 2 e. CC /\ 2 =/= 0 ) ) |
10 |
|
divdir |
|- ( ( ( N - 1 ) e. CC /\ 1 e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( ( N - 1 ) + 1 ) / 2 ) = ( ( ( N - 1 ) / 2 ) + ( 1 / 2 ) ) ) |
11 |
6 7 9 10
|
syl3anc |
|- ( N e. CC -> ( ( ( N - 1 ) + 1 ) / 2 ) = ( ( ( N - 1 ) / 2 ) + ( 1 / 2 ) ) ) |
12 |
5 11
|
eqtrd |
|- ( N e. CC -> ( N / 2 ) = ( ( ( N - 1 ) / 2 ) + ( 1 / 2 ) ) ) |
13 |
2 12
|
syl |
|- ( N e. Odd -> ( N / 2 ) = ( ( ( N - 1 ) / 2 ) + ( 1 / 2 ) ) ) |
14 |
13
|
fveq2d |
|- ( N e. Odd -> ( |_ ` ( N / 2 ) ) = ( |_ ` ( ( ( N - 1 ) / 2 ) + ( 1 / 2 ) ) ) ) |
15 |
|
halfge0 |
|- 0 <_ ( 1 / 2 ) |
16 |
|
halflt1 |
|- ( 1 / 2 ) < 1 |
17 |
15 16
|
pm3.2i |
|- ( 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) < 1 ) |
18 |
|
oddm1div2z |
|- ( N e. Odd -> ( ( N - 1 ) / 2 ) e. ZZ ) |
19 |
|
halfre |
|- ( 1 / 2 ) e. RR |
20 |
|
flbi2 |
|- ( ( ( ( N - 1 ) / 2 ) e. ZZ /\ ( 1 / 2 ) e. RR ) -> ( ( |_ ` ( ( ( N - 1 ) / 2 ) + ( 1 / 2 ) ) ) = ( ( N - 1 ) / 2 ) <-> ( 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) < 1 ) ) ) |
21 |
18 19 20
|
sylancl |
|- ( N e. Odd -> ( ( |_ ` ( ( ( N - 1 ) / 2 ) + ( 1 / 2 ) ) ) = ( ( N - 1 ) / 2 ) <-> ( 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) < 1 ) ) ) |
22 |
17 21
|
mpbiri |
|- ( N e. Odd -> ( |_ ` ( ( ( N - 1 ) / 2 ) + ( 1 / 2 ) ) ) = ( ( N - 1 ) / 2 ) ) |
23 |
14 22
|
eqtrd |
|- ( N e. Odd -> ( |_ ` ( N / 2 ) ) = ( ( N - 1 ) / 2 ) ) |