Step |
Hyp |
Ref |
Expression |
1 |
|
id |
|- ( A e. ZZ -> A e. ZZ ) |
2 |
|
oveq2 |
|- ( k = A -> ( 2 x. k ) = ( 2 x. A ) ) |
3 |
2
|
oveq1d |
|- ( k = A -> ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) ) |
4 |
3
|
eqeq1d |
|- ( k = A -> ( ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) <-> ( ( 2 x. A ) + 1 ) = ( ( 2 x. A ) + 1 ) ) ) |
5 |
4
|
adantl |
|- ( ( A e. ZZ /\ k = A ) -> ( ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) <-> ( ( 2 x. A ) + 1 ) = ( ( 2 x. A ) + 1 ) ) ) |
6 |
|
eqidd |
|- ( A e. ZZ -> ( ( 2 x. A ) + 1 ) = ( ( 2 x. A ) + 1 ) ) |
7 |
1 5 6
|
rspcedvd |
|- ( A e. ZZ -> E. k e. ZZ ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) ) |
8 |
|
2z |
|- 2 e. ZZ |
9 |
8
|
a1i |
|- ( A e. ZZ -> 2 e. ZZ ) |
10 |
9 1
|
zmulcld |
|- ( A e. ZZ -> ( 2 x. A ) e. ZZ ) |
11 |
10
|
peano2zd |
|- ( A e. ZZ -> ( ( 2 x. A ) + 1 ) e. ZZ ) |
12 |
|
odd2np1 |
|- ( ( ( 2 x. A ) + 1 ) e. ZZ -> ( -. 2 || ( ( 2 x. A ) + 1 ) <-> E. k e. ZZ ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) ) ) |
13 |
11 12
|
syl |
|- ( A e. ZZ -> ( -. 2 || ( ( 2 x. A ) + 1 ) <-> E. k e. ZZ ( ( 2 x. k ) + 1 ) = ( ( 2 x. A ) + 1 ) ) ) |
14 |
7 13
|
mpbird |
|- ( A e. ZZ -> -. 2 || ( ( 2 x. A ) + 1 ) ) |
15 |
14
|
adantr |
|- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || ( ( 2 x. A ) + 1 ) ) |
16 |
|
breq2 |
|- ( B = ( ( 2 x. A ) + 1 ) -> ( 2 || B <-> 2 || ( ( 2 x. A ) + 1 ) ) ) |
17 |
16
|
adantl |
|- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> ( 2 || B <-> 2 || ( ( 2 x. A ) + 1 ) ) ) |
18 |
15 17
|
mtbird |
|- ( ( A e. ZZ /\ B = ( ( 2 x. A ) + 1 ) ) -> -. 2 || B ) |