Step |
Hyp |
Ref |
Expression |
1 |
|
zlmodzxzldep.z |
|- Z = ( ZZring freeLMod { 0 , 1 } ) |
2 |
|
zlmodzxzldep.a |
|- A = { <. 0 , 3 >. , <. 1 , 6 >. } |
3 |
|
zlmodzxzldep.b |
|- B = { <. 0 , 2 >. , <. 1 , 4 >. } |
4 |
|
zlmodzxzldeplem.f |
|- F = { <. A , 2 >. , <. B , -u 3 >. } |
5 |
|
prex |
|- { <. 0 , 3 >. , <. 1 , 6 >. } e. _V |
6 |
2 5
|
eqeltri |
|- A e. _V |
7 |
|
prex |
|- { <. 0 , 2 >. , <. 1 , 4 >. } e. _V |
8 |
3 7
|
eqeltri |
|- B e. _V |
9 |
|
2ne0 |
|- 2 =/= 0 |
10 |
4
|
fveq1i |
|- ( F ` A ) = ( { <. A , 2 >. , <. B , -u 3 >. } ` A ) |
11 |
1 2 3
|
zlmodzxzldeplem |
|- A =/= B |
12 |
|
2ex |
|- 2 e. _V |
13 |
6 12
|
fvpr1 |
|- ( A =/= B -> ( { <. A , 2 >. , <. B , -u 3 >. } ` A ) = 2 ) |
14 |
11 13
|
mp1i |
|- ( ( A e. _V /\ B e. _V ) -> ( { <. A , 2 >. , <. B , -u 3 >. } ` A ) = 2 ) |
15 |
10 14
|
syl5eq |
|- ( ( A e. _V /\ B e. _V ) -> ( F ` A ) = 2 ) |
16 |
15
|
neeq1d |
|- ( ( A e. _V /\ B e. _V ) -> ( ( F ` A ) =/= 0 <-> 2 =/= 0 ) ) |
17 |
9 16
|
mpbiri |
|- ( ( A e. _V /\ B e. _V ) -> ( F ` A ) =/= 0 ) |
18 |
17
|
orcd |
|- ( ( A e. _V /\ B e. _V ) -> ( ( F ` A ) =/= 0 \/ ( F ` B ) =/= 0 ) ) |
19 |
|
fveq2 |
|- ( y = A -> ( F ` y ) = ( F ` A ) ) |
20 |
19
|
neeq1d |
|- ( y = A -> ( ( F ` y ) =/= 0 <-> ( F ` A ) =/= 0 ) ) |
21 |
|
fveq2 |
|- ( y = B -> ( F ` y ) = ( F ` B ) ) |
22 |
21
|
neeq1d |
|- ( y = B -> ( ( F ` y ) =/= 0 <-> ( F ` B ) =/= 0 ) ) |
23 |
20 22
|
rexprg |
|- ( ( A e. _V /\ B e. _V ) -> ( E. y e. { A , B } ( F ` y ) =/= 0 <-> ( ( F ` A ) =/= 0 \/ ( F ` B ) =/= 0 ) ) ) |
24 |
18 23
|
mpbird |
|- ( ( A e. _V /\ B e. _V ) -> E. y e. { A , B } ( F ` y ) =/= 0 ) |
25 |
6 8 24
|
mp2an |
|- E. y e. { A , B } ( F ` y ) =/= 0 |