Step |
Hyp |
Ref |
Expression |
1 |
|
lsmsnpridl.1 |
|- B = ( Base ` R ) |
2 |
|
lsmsnpridl.2 |
|- G = ( mulGrp ` R ) |
3 |
|
lsmsnpridl.3 |
|- .X. = ( LSSum ` G ) |
4 |
|
lsmsnpridl.4 |
|- K = ( RSpan ` R ) |
5 |
|
lsmsnpridl.5 |
|- ( ph -> R e. Ring ) |
6 |
|
lsmsnpridl.6 |
|- ( ph -> X e. B ) |
7 |
2 1
|
mgpbas |
|- B = ( Base ` G ) |
8 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
9 |
2 8
|
mgpplusg |
|- ( .r ` R ) = ( +g ` G ) |
10 |
2
|
fvexi |
|- G e. _V |
11 |
10
|
a1i |
|- ( ph -> G e. _V ) |
12 |
|
ssidd |
|- ( ph -> B C_ B ) |
13 |
7 9 3 11 12 6
|
elgrplsmsn |
|- ( ph -> ( x e. ( B .X. { X } ) <-> E. y e. B x = ( y ( .r ` R ) X ) ) ) |
14 |
1 8 4
|
rspsnel |
|- ( ( R e. Ring /\ X e. B ) -> ( x e. ( K ` { X } ) <-> E. y e. B x = ( y ( .r ` R ) X ) ) ) |
15 |
5 6 14
|
syl2anc |
|- ( ph -> ( x e. ( K ` { X } ) <-> E. y e. B x = ( y ( .r ` R ) X ) ) ) |
16 |
13 15
|
bitr4d |
|- ( ph -> ( x e. ( B .X. { X } ) <-> x e. ( K ` { X } ) ) ) |
17 |
16
|
eqrdv |
|- ( ph -> ( B .X. { X } ) = ( K ` { X } ) ) |