Step |
Hyp |
Ref |
Expression |
1 |
|
2zrng.e |
|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) } |
2 |
|
2z |
|- 2 e. ZZ |
3 |
|
2cn |
|- 2 e. CC |
4 |
|
1zzd |
|- ( 2 e. CC -> 1 e. ZZ ) |
5 |
|
oveq2 |
|- ( x = 1 -> ( 2 x. x ) = ( 2 x. 1 ) ) |
6 |
5
|
eqeq2d |
|- ( x = 1 -> ( 2 = ( 2 x. x ) <-> 2 = ( 2 x. 1 ) ) ) |
7 |
6
|
adantl |
|- ( ( 2 e. CC /\ x = 1 ) -> ( 2 = ( 2 x. x ) <-> 2 = ( 2 x. 1 ) ) ) |
8 |
|
mulid1 |
|- ( 2 e. CC -> ( 2 x. 1 ) = 2 ) |
9 |
8
|
eqcomd |
|- ( 2 e. CC -> 2 = ( 2 x. 1 ) ) |
10 |
4 7 9
|
rspcedvd |
|- ( 2 e. CC -> E. x e. ZZ 2 = ( 2 x. x ) ) |
11 |
3 10
|
ax-mp |
|- E. x e. ZZ 2 = ( 2 x. x ) |
12 |
|
eqeq1 |
|- ( z = 2 -> ( z = ( 2 x. x ) <-> 2 = ( 2 x. x ) ) ) |
13 |
12
|
rexbidv |
|- ( z = 2 -> ( E. x e. ZZ z = ( 2 x. x ) <-> E. x e. ZZ 2 = ( 2 x. x ) ) ) |
14 |
13
|
elrab |
|- ( 2 e. { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) } <-> ( 2 e. ZZ /\ E. x e. ZZ 2 = ( 2 x. x ) ) ) |
15 |
2 11 14
|
mpbir2an |
|- 2 e. { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) } |
16 |
15 1
|
eleqtrri |
|- 2 e. E |