Step |
Hyp |
Ref |
Expression |
1 |
|
dsmmcl.p |
|- P = ( S Xs_ R ) |
2 |
|
dsmmcl.h |
|- H = ( Base ` ( S (+)m R ) ) |
3 |
|
dsmmcl.i |
|- ( ph -> I e. W ) |
4 |
|
dsmmcl.s |
|- ( ph -> S e. V ) |
5 |
|
dsmmcl.r |
|- ( ph -> R : I --> Mnd ) |
6 |
|
dsmm0cl.z |
|- .0. = ( 0g ` P ) |
7 |
1 3 4 5
|
prdsmndd |
|- ( ph -> P e. Mnd ) |
8 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
9 |
8 6
|
mndidcl |
|- ( P e. Mnd -> .0. e. ( Base ` P ) ) |
10 |
7 9
|
syl |
|- ( ph -> .0. e. ( Base ` P ) ) |
11 |
1 3 4 5
|
prds0g |
|- ( ph -> ( 0g o. R ) = ( 0g ` P ) ) |
12 |
11 6
|
eqtr4di |
|- ( ph -> ( 0g o. R ) = .0. ) |
13 |
12
|
adantr |
|- ( ( ph /\ a e. I ) -> ( 0g o. R ) = .0. ) |
14 |
13
|
fveq1d |
|- ( ( ph /\ a e. I ) -> ( ( 0g o. R ) ` a ) = ( .0. ` a ) ) |
15 |
5
|
ffnd |
|- ( ph -> R Fn I ) |
16 |
|
fvco2 |
|- ( ( R Fn I /\ a e. I ) -> ( ( 0g o. R ) ` a ) = ( 0g ` ( R ` a ) ) ) |
17 |
15 16
|
sylan |
|- ( ( ph /\ a e. I ) -> ( ( 0g o. R ) ` a ) = ( 0g ` ( R ` a ) ) ) |
18 |
14 17
|
eqtr3d |
|- ( ( ph /\ a e. I ) -> ( .0. ` a ) = ( 0g ` ( R ` a ) ) ) |
19 |
|
nne |
|- ( -. ( .0. ` a ) =/= ( 0g ` ( R ` a ) ) <-> ( .0. ` a ) = ( 0g ` ( R ` a ) ) ) |
20 |
18 19
|
sylibr |
|- ( ( ph /\ a e. I ) -> -. ( .0. ` a ) =/= ( 0g ` ( R ` a ) ) ) |
21 |
20
|
ralrimiva |
|- ( ph -> A. a e. I -. ( .0. ` a ) =/= ( 0g ` ( R ` a ) ) ) |
22 |
|
rabeq0 |
|- ( { a e. I | ( .0. ` a ) =/= ( 0g ` ( R ` a ) ) } = (/) <-> A. a e. I -. ( .0. ` a ) =/= ( 0g ` ( R ` a ) ) ) |
23 |
21 22
|
sylibr |
|- ( ph -> { a e. I | ( .0. ` a ) =/= ( 0g ` ( R ` a ) ) } = (/) ) |
24 |
|
0fin |
|- (/) e. Fin |
25 |
23 24
|
eqeltrdi |
|- ( ph -> { a e. I | ( .0. ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin ) |
26 |
|
eqid |
|- ( S (+)m R ) = ( S (+)m R ) |
27 |
1 26 8 2 3 15
|
dsmmelbas |
|- ( ph -> ( .0. e. H <-> ( .0. e. ( Base ` P ) /\ { a e. I | ( .0. ` a ) =/= ( 0g ` ( R ` a ) ) } e. Fin ) ) ) |
28 |
10 25 27
|
mpbir2and |
|- ( ph -> .0. e. H ) |