Step |
Hyp |
Ref |
Expression |
1 |
|
lpival.p |
|- P = ( LPIdeal ` R ) |
2 |
|
lpival.k |
|- K = ( RSpan ` R ) |
3 |
|
lpival.b |
|- B = ( Base ` R ) |
4 |
|
fveq2 |
|- ( r = R -> ( Base ` r ) = ( Base ` R ) ) |
5 |
|
fveq2 |
|- ( r = R -> ( RSpan ` r ) = ( RSpan ` R ) ) |
6 |
5
|
fveq1d |
|- ( r = R -> ( ( RSpan ` r ) ` { g } ) = ( ( RSpan ` R ) ` { g } ) ) |
7 |
6
|
sneqd |
|- ( r = R -> { ( ( RSpan ` r ) ` { g } ) } = { ( ( RSpan ` R ) ` { g } ) } ) |
8 |
4 7
|
iuneq12d |
|- ( r = R -> U_ g e. ( Base ` r ) { ( ( RSpan ` r ) ` { g } ) } = U_ g e. ( Base ` R ) { ( ( RSpan ` R ) ` { g } ) } ) |
9 |
|
df-lpidl |
|- LPIdeal = ( r e. Ring |-> U_ g e. ( Base ` r ) { ( ( RSpan ` r ) ` { g } ) } ) |
10 |
|
fvex |
|- ( RSpan ` R ) e. _V |
11 |
10
|
rnex |
|- ran ( RSpan ` R ) e. _V |
12 |
|
p0ex |
|- { (/) } e. _V |
13 |
11 12
|
unex |
|- ( ran ( RSpan ` R ) u. { (/) } ) e. _V |
14 |
|
iunss |
|- ( U_ g e. ( Base ` R ) { ( ( RSpan ` R ) ` { g } ) } C_ ( ran ( RSpan ` R ) u. { (/) } ) <-> A. g e. ( Base ` R ) { ( ( RSpan ` R ) ` { g } ) } C_ ( ran ( RSpan ` R ) u. { (/) } ) ) |
15 |
|
fvrn0 |
|- ( ( RSpan ` R ) ` { g } ) e. ( ran ( RSpan ` R ) u. { (/) } ) |
16 |
|
snssi |
|- ( ( ( RSpan ` R ) ` { g } ) e. ( ran ( RSpan ` R ) u. { (/) } ) -> { ( ( RSpan ` R ) ` { g } ) } C_ ( ran ( RSpan ` R ) u. { (/) } ) ) |
17 |
15 16
|
ax-mp |
|- { ( ( RSpan ` R ) ` { g } ) } C_ ( ran ( RSpan ` R ) u. { (/) } ) |
18 |
17
|
a1i |
|- ( g e. ( Base ` R ) -> { ( ( RSpan ` R ) ` { g } ) } C_ ( ran ( RSpan ` R ) u. { (/) } ) ) |
19 |
14 18
|
mprgbir |
|- U_ g e. ( Base ` R ) { ( ( RSpan ` R ) ` { g } ) } C_ ( ran ( RSpan ` R ) u. { (/) } ) |
20 |
13 19
|
ssexi |
|- U_ g e. ( Base ` R ) { ( ( RSpan ` R ) ` { g } ) } e. _V |
21 |
8 9 20
|
fvmpt |
|- ( R e. Ring -> ( LPIdeal ` R ) = U_ g e. ( Base ` R ) { ( ( RSpan ` R ) ` { g } ) } ) |
22 |
|
iuneq1 |
|- ( B = ( Base ` R ) -> U_ g e. B { ( K ` { g } ) } = U_ g e. ( Base ` R ) { ( K ` { g } ) } ) |
23 |
3 22
|
ax-mp |
|- U_ g e. B { ( K ` { g } ) } = U_ g e. ( Base ` R ) { ( K ` { g } ) } |
24 |
2
|
fveq1i |
|- ( K ` { g } ) = ( ( RSpan ` R ) ` { g } ) |
25 |
24
|
sneqi |
|- { ( K ` { g } ) } = { ( ( RSpan ` R ) ` { g } ) } |
26 |
25
|
a1i |
|- ( g e. ( Base ` R ) -> { ( K ` { g } ) } = { ( ( RSpan ` R ) ` { g } ) } ) |
27 |
26
|
iuneq2i |
|- U_ g e. ( Base ` R ) { ( K ` { g } ) } = U_ g e. ( Base ` R ) { ( ( RSpan ` R ) ` { g } ) } |
28 |
23 27
|
eqtri |
|- U_ g e. B { ( K ` { g } ) } = U_ g e. ( Base ` R ) { ( ( RSpan ` R ) ` { g } ) } |
29 |
21 1 28
|
3eqtr4g |
|- ( R e. Ring -> P = U_ g e. B { ( K ` { g } ) } ) |