Step |
Hyp |
Ref |
Expression |
1 |
|
dihval.b |
|- B = ( Base ` K ) |
2 |
|
dihval.l |
|- .<_ = ( le ` K ) |
3 |
|
dihval.j |
|- .\/ = ( join ` K ) |
4 |
|
dihval.m |
|- ./\ = ( meet ` K ) |
5 |
|
dihval.a |
|- A = ( Atoms ` K ) |
6 |
|
dihval.h |
|- H = ( LHyp ` K ) |
7 |
|
dihval.i |
|- I = ( ( DIsoH ` K ) ` W ) |
8 |
|
dihval.d |
|- D = ( ( DIsoB ` K ) ` W ) |
9 |
|
dihval.c |
|- C = ( ( DIsoC ` K ) ` W ) |
10 |
|
dihval.u |
|- U = ( ( DVecH ` K ) ` W ) |
11 |
|
dihval.s |
|- S = ( LSubSp ` U ) |
12 |
|
dihval.p |
|- .(+) = ( LSSum ` U ) |
13 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dihval |
|- ( ( ( K e. V /\ W e. H ) /\ X e. B ) -> ( I ` X ) = if ( X .<_ W , ( D ` X ) , ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) ) ) |
14 |
|
iffalse |
|- ( -. X .<_ W -> if ( X .<_ W , ( D ` X ) , ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) ) = ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) ) |
15 |
13 14
|
sylan9eq |
|- ( ( ( ( K e. V /\ W e. H ) /\ X e. B ) /\ -. X .<_ W ) -> ( I ` X ) = ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) ) |
16 |
15
|
anasss |
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( I ` X ) = ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) ) |