Step |
Hyp |
Ref |
Expression |
1 |
|
cntrval.b |
|- B = ( Base ` M ) |
2 |
|
cntrval.z |
|- Z = ( Cntz ` M ) |
3 |
|
fveq2 |
|- ( m = M -> ( Cntz ` m ) = ( Cntz ` M ) ) |
4 |
3 2
|
eqtr4di |
|- ( m = M -> ( Cntz ` m ) = Z ) |
5 |
|
fveq2 |
|- ( m = M -> ( Base ` m ) = ( Base ` M ) ) |
6 |
5 1
|
eqtr4di |
|- ( m = M -> ( Base ` m ) = B ) |
7 |
4 6
|
fveq12d |
|- ( m = M -> ( ( Cntz ` m ) ` ( Base ` m ) ) = ( Z ` B ) ) |
8 |
|
df-cntr |
|- Cntr = ( m e. _V |-> ( ( Cntz ` m ) ` ( Base ` m ) ) ) |
9 |
|
fvex |
|- ( Z ` B ) e. _V |
10 |
7 8 9
|
fvmpt |
|- ( M e. _V -> ( Cntr ` M ) = ( Z ` B ) ) |
11 |
10
|
eqcomd |
|- ( M e. _V -> ( Z ` B ) = ( Cntr ` M ) ) |
12 |
|
0fv |
|- ( (/) ` B ) = (/) |
13 |
|
fvprc |
|- ( -. M e. _V -> ( Cntz ` M ) = (/) ) |
14 |
2 13
|
syl5eq |
|- ( -. M e. _V -> Z = (/) ) |
15 |
14
|
fveq1d |
|- ( -. M e. _V -> ( Z ` B ) = ( (/) ` B ) ) |
16 |
|
fvprc |
|- ( -. M e. _V -> ( Cntr ` M ) = (/) ) |
17 |
12 15 16
|
3eqtr4a |
|- ( -. M e. _V -> ( Z ` B ) = ( Cntr ` M ) ) |
18 |
11 17
|
pm2.61i |
|- ( Z ` B ) = ( Cntr ` M ) |