Step |
Hyp |
Ref |
Expression |
1 |
|
dihjust.b |
|- B = ( Base ` K ) |
2 |
|
dihjust.l |
|- .<_ = ( le ` K ) |
3 |
|
dihjust.j |
|- .\/ = ( join ` K ) |
4 |
|
dihjust.m |
|- ./\ = ( meet ` K ) |
5 |
|
dihjust.a |
|- A = ( Atoms ` K ) |
6 |
|
dihjust.h |
|- H = ( LHyp ` K ) |
7 |
|
dihjust.i |
|- I = ( ( DIsoB ` K ) ` W ) |
8 |
|
dihjust.J |
|- J = ( ( DIsoC ` K ) ` W ) |
9 |
|
dihjust.u |
|- U = ( ( DVecH ` K ) ` W ) |
10 |
|
dihjust.s |
|- .(+) = ( LSSum ` U ) |
11 |
1 2 3 4 5 6 7 8 9 10
|
dihjustlem |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) ) |
12 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( K e. HL /\ W e. H ) ) |
13 |
|
simp22 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( R e. A /\ -. R .<_ W ) ) |
14 |
|
simp21 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
15 |
|
simp23 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> X e. B ) |
16 |
|
simp3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) |
17 |
16
|
eqcomd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( R .\/ ( X ./\ W ) ) = ( Q .\/ ( X ./\ W ) ) ) |
18 |
1 2 3 4 5 6 7 8 9 10
|
dihjustlem |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ X e. B ) /\ ( R .\/ ( X ./\ W ) ) = ( Q .\/ ( X ./\ W ) ) ) -> ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) ) |
19 |
12 13 14 15 17 18
|
syl131anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) C_ ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) ) |
20 |
11 19
|
eqssd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ X e. B ) /\ ( Q .\/ ( X ./\ W ) ) = ( R .\/ ( X ./\ W ) ) ) -> ( ( J ` Q ) .(+) ( I ` ( X ./\ W ) ) ) = ( ( J ` R ) .(+) ( I ` ( X ./\ W ) ) ) ) |