Step |
Hyp |
Ref |
Expression |
1 |
|
ellspds.n |
|- N = ( LSpan ` M ) |
2 |
|
ellspds.v |
|- B = ( Base ` M ) |
3 |
|
ellspds.k |
|- K = ( Base ` S ) |
4 |
|
ellspds.s |
|- S = ( Scalar ` M ) |
5 |
|
ellspds.z |
|- .0. = ( 0g ` S ) |
6 |
|
ellspds.t |
|- .x. = ( .s ` M ) |
7 |
|
ellspds.m |
|- ( ph -> M e. LMod ) |
8 |
|
ellspds.1 |
|- ( ph -> V C_ B ) |
9 |
|
f1oi |
|- ( _I |` V ) : V -1-1-onto-> V |
10 |
|
f1of |
|- ( ( _I |` V ) : V -1-1-onto-> V -> ( _I |` V ) : V --> V ) |
11 |
9 10
|
mp1i |
|- ( ph -> ( _I |` V ) : V --> V ) |
12 |
11 8
|
fssd |
|- ( ph -> ( _I |` V ) : V --> B ) |
13 |
2
|
fvexi |
|- B e. _V |
14 |
13
|
a1i |
|- ( ph -> B e. _V ) |
15 |
14 8
|
ssexd |
|- ( ph -> V e. _V ) |
16 |
1 2 3 4 5 6 12 7 15
|
ellspd |
|- ( ph -> ( X e. ( N ` ( ( _I |` V ) " V ) ) <-> E. a e. ( K ^m V ) ( a finSupp .0. /\ X = ( M gsum ( a oF .x. ( _I |` V ) ) ) ) ) ) |
17 |
|
ssid |
|- V C_ V |
18 |
|
resiima |
|- ( V C_ V -> ( ( _I |` V ) " V ) = V ) |
19 |
17 18
|
mp1i |
|- ( ph -> ( ( _I |` V ) " V ) = V ) |
20 |
19
|
fveq2d |
|- ( ph -> ( N ` ( ( _I |` V ) " V ) ) = ( N ` V ) ) |
21 |
20
|
eleq2d |
|- ( ph -> ( X e. ( N ` ( ( _I |` V ) " V ) ) <-> X e. ( N ` V ) ) ) |
22 |
|
elmapfn |
|- ( a e. ( K ^m V ) -> a Fn V ) |
23 |
22
|
adantl |
|- ( ( ph /\ a e. ( K ^m V ) ) -> a Fn V ) |
24 |
9 10
|
mp1i |
|- ( ( ph /\ a e. ( K ^m V ) ) -> ( _I |` V ) : V --> V ) |
25 |
24
|
ffnd |
|- ( ( ph /\ a e. ( K ^m V ) ) -> ( _I |` V ) Fn V ) |
26 |
15
|
adantr |
|- ( ( ph /\ a e. ( K ^m V ) ) -> V e. _V ) |
27 |
|
inidm |
|- ( V i^i V ) = V |
28 |
|
eqidd |
|- ( ( ( ph /\ a e. ( K ^m V ) ) /\ v e. V ) -> ( a ` v ) = ( a ` v ) ) |
29 |
|
fvresi |
|- ( v e. V -> ( ( _I |` V ) ` v ) = v ) |
30 |
29
|
adantl |
|- ( ( ( ph /\ a e. ( K ^m V ) ) /\ v e. V ) -> ( ( _I |` V ) ` v ) = v ) |
31 |
23 25 26 26 27 28 30
|
offval |
|- ( ( ph /\ a e. ( K ^m V ) ) -> ( a oF .x. ( _I |` V ) ) = ( v e. V |-> ( ( a ` v ) .x. v ) ) ) |
32 |
31
|
oveq2d |
|- ( ( ph /\ a e. ( K ^m V ) ) -> ( M gsum ( a oF .x. ( _I |` V ) ) ) = ( M gsum ( v e. V |-> ( ( a ` v ) .x. v ) ) ) ) |
33 |
32
|
eqeq2d |
|- ( ( ph /\ a e. ( K ^m V ) ) -> ( X = ( M gsum ( a oF .x. ( _I |` V ) ) ) <-> X = ( M gsum ( v e. V |-> ( ( a ` v ) .x. v ) ) ) ) ) |
34 |
33
|
anbi2d |
|- ( ( ph /\ a e. ( K ^m V ) ) -> ( ( a finSupp .0. /\ X = ( M gsum ( a oF .x. ( _I |` V ) ) ) ) <-> ( a finSupp .0. /\ X = ( M gsum ( v e. V |-> ( ( a ` v ) .x. v ) ) ) ) ) ) |
35 |
34
|
rexbidva |
|- ( ph -> ( E. a e. ( K ^m V ) ( a finSupp .0. /\ X = ( M gsum ( a oF .x. ( _I |` V ) ) ) ) <-> E. a e. ( K ^m V ) ( a finSupp .0. /\ X = ( M gsum ( v e. V |-> ( ( a ` v ) .x. v ) ) ) ) ) ) |
36 |
16 21 35
|
3bitr3d |
|- ( ph -> ( X e. ( N ` V ) <-> E. a e. ( K ^m V ) ( a finSupp .0. /\ X = ( M gsum ( v e. V |-> ( ( a ` v ) .x. v ) ) ) ) ) ) |