Step |
Hyp |
Ref |
Expression |
1 |
|
ulmdv.z |
|- Z = ( ZZ>= ` M ) |
2 |
|
ulmdv.s |
|- ( ph -> S e. { RR , CC } ) |
3 |
|
ulmdv.m |
|- ( ph -> M e. ZZ ) |
4 |
|
ulmdv.f |
|- ( ph -> F : Z --> ( CC ^m X ) ) |
5 |
|
ulmdv.g |
|- ( ph -> G : X --> CC ) |
6 |
|
ulmdv.l |
|- ( ( ph /\ z e. X ) -> ( k e. Z |-> ( ( F ` k ) ` z ) ) ~~> ( G ` z ) ) |
7 |
|
ulmdv.u |
|- ( ph -> ( k e. Z |-> ( S _D ( F ` k ) ) ) ( ~~>u ` X ) H ) |
8 |
|
ovex |
|- ( S _D ( F ` k ) ) e. _V |
9 |
8
|
rgenw |
|- A. k e. Z ( S _D ( F ` k ) ) e. _V |
10 |
|
eqid |
|- ( k e. Z |-> ( S _D ( F ` k ) ) ) = ( k e. Z |-> ( S _D ( F ` k ) ) ) |
11 |
10
|
fnmpt |
|- ( A. k e. Z ( S _D ( F ` k ) ) e. _V -> ( k e. Z |-> ( S _D ( F ` k ) ) ) Fn Z ) |
12 |
9 11
|
mp1i |
|- ( ph -> ( k e. Z |-> ( S _D ( F ` k ) ) ) Fn Z ) |
13 |
|
ulmf2 |
|- ( ( ( k e. Z |-> ( S _D ( F ` k ) ) ) Fn Z /\ ( k e. Z |-> ( S _D ( F ` k ) ) ) ( ~~>u ` X ) H ) -> ( k e. Z |-> ( S _D ( F ` k ) ) ) : Z --> ( CC ^m X ) ) |
14 |
12 7 13
|
syl2anc |
|- ( ph -> ( k e. Z |-> ( S _D ( F ` k ) ) ) : Z --> ( CC ^m X ) ) |
15 |
14
|
fvmptelrn |
|- ( ( ph /\ k e. Z ) -> ( S _D ( F ` k ) ) e. ( CC ^m X ) ) |
16 |
|
elmapi |
|- ( ( S _D ( F ` k ) ) e. ( CC ^m X ) -> ( S _D ( F ` k ) ) : X --> CC ) |
17 |
|
fdm |
|- ( ( S _D ( F ` k ) ) : X --> CC -> dom ( S _D ( F ` k ) ) = X ) |
18 |
15 16 17
|
3syl |
|- ( ( ph /\ k e. Z ) -> dom ( S _D ( F ` k ) ) = X ) |