Step |
Hyp |
Ref |
Expression |
1 |
|
reprval.a |
|- ( ph -> A C_ NN ) |
2 |
|
reprval.m |
|- ( ph -> M e. ZZ ) |
3 |
|
reprval.s |
|- ( ph -> S e. NN0 ) |
4 |
|
reprfi.1 |
|- ( ph -> A e. Fin ) |
5 |
1 2 3
|
reprval |
|- ( ph -> ( A ( repr ` S ) M ) = { c e. ( A ^m ( 0 ..^ S ) ) | sum_ a e. ( 0 ..^ S ) ( c ` a ) = M } ) |
6 |
|
fzofi |
|- ( 0 ..^ S ) e. Fin |
7 |
|
mapfi |
|- ( ( A e. Fin /\ ( 0 ..^ S ) e. Fin ) -> ( A ^m ( 0 ..^ S ) ) e. Fin ) |
8 |
4 6 7
|
sylancl |
|- ( ph -> ( A ^m ( 0 ..^ S ) ) e. Fin ) |
9 |
|
rabfi |
|- ( ( A ^m ( 0 ..^ S ) ) e. Fin -> { c e. ( A ^m ( 0 ..^ S ) ) | sum_ a e. ( 0 ..^ S ) ( c ` a ) = M } e. Fin ) |
10 |
8 9
|
syl |
|- ( ph -> { c e. ( A ^m ( 0 ..^ S ) ) | sum_ a e. ( 0 ..^ S ) ( c ` a ) = M } e. Fin ) |
11 |
5 10
|
eqeltrd |
|- ( ph -> ( A ( repr ` S ) M ) e. Fin ) |