Step |
Hyp |
Ref |
Expression |
1 |
|
0nn0 |
|- 0 e. NN0 |
2 |
|
bitsval2 |
|- ( ( N e. ZZ /\ 0 e. NN0 ) -> ( 0 e. ( bits ` N ) <-> -. 2 || ( |_ ` ( N / ( 2 ^ 0 ) ) ) ) ) |
3 |
1 2
|
mpan2 |
|- ( N e. ZZ -> ( 0 e. ( bits ` N ) <-> -. 2 || ( |_ ` ( N / ( 2 ^ 0 ) ) ) ) ) |
4 |
|
2cn |
|- 2 e. CC |
5 |
|
exp0 |
|- ( 2 e. CC -> ( 2 ^ 0 ) = 1 ) |
6 |
4 5
|
ax-mp |
|- ( 2 ^ 0 ) = 1 |
7 |
6
|
oveq2i |
|- ( N / ( 2 ^ 0 ) ) = ( N / 1 ) |
8 |
|
zcn |
|- ( N e. ZZ -> N e. CC ) |
9 |
8
|
div1d |
|- ( N e. ZZ -> ( N / 1 ) = N ) |
10 |
7 9
|
syl5eq |
|- ( N e. ZZ -> ( N / ( 2 ^ 0 ) ) = N ) |
11 |
10
|
fveq2d |
|- ( N e. ZZ -> ( |_ ` ( N / ( 2 ^ 0 ) ) ) = ( |_ ` N ) ) |
12 |
|
flid |
|- ( N e. ZZ -> ( |_ ` N ) = N ) |
13 |
11 12
|
eqtrd |
|- ( N e. ZZ -> ( |_ ` ( N / ( 2 ^ 0 ) ) ) = N ) |
14 |
13
|
breq2d |
|- ( N e. ZZ -> ( 2 || ( |_ ` ( N / ( 2 ^ 0 ) ) ) <-> 2 || N ) ) |
15 |
14
|
notbid |
|- ( N e. ZZ -> ( -. 2 || ( |_ ` ( N / ( 2 ^ 0 ) ) ) <-> -. 2 || N ) ) |
16 |
|
isodd3 |
|- ( N e. Odd <-> ( N e. ZZ /\ -. 2 || N ) ) |
17 |
16
|
baibr |
|- ( N e. ZZ -> ( -. 2 || N <-> N e. Odd ) ) |
18 |
3 15 17
|
3bitrd |
|- ( N e. ZZ -> ( 0 e. ( bits ` N ) <-> N e. Odd ) ) |