Step |
Hyp |
Ref |
Expression |
1 |
|
lpival.p |
|- P = ( LPIdeal ` R ) |
2 |
|
lpi1.b |
|- B = ( Base ` R ) |
3 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
4 |
2 3
|
ringidcl |
|- ( R e. Ring -> ( 1r ` R ) e. B ) |
5 |
|
eqid |
|- ( RSpan ` R ) = ( RSpan ` R ) |
6 |
5 2 3
|
rsp1 |
|- ( R e. Ring -> ( ( RSpan ` R ) ` { ( 1r ` R ) } ) = B ) |
7 |
6
|
eqcomd |
|- ( R e. Ring -> B = ( ( RSpan ` R ) ` { ( 1r ` R ) } ) ) |
8 |
|
sneq |
|- ( g = ( 1r ` R ) -> { g } = { ( 1r ` R ) } ) |
9 |
8
|
fveq2d |
|- ( g = ( 1r ` R ) -> ( ( RSpan ` R ) ` { g } ) = ( ( RSpan ` R ) ` { ( 1r ` R ) } ) ) |
10 |
9
|
rspceeqv |
|- ( ( ( 1r ` R ) e. B /\ B = ( ( RSpan ` R ) ` { ( 1r ` R ) } ) ) -> E. g e. B B = ( ( RSpan ` R ) ` { g } ) ) |
11 |
4 7 10
|
syl2anc |
|- ( R e. Ring -> E. g e. B B = ( ( RSpan ` R ) ` { g } ) ) |
12 |
1 5 2
|
islpidl |
|- ( R e. Ring -> ( B e. P <-> E. g e. B B = ( ( RSpan ` R ) ` { g } ) ) ) |
13 |
11 12
|
mpbird |
|- ( R e. Ring -> B e. P ) |