Step |
Hyp |
Ref |
Expression |
1 |
|
4that.l |
|- .<_ = ( le ` K ) |
2 |
|
4that.j |
|- .\/ = ( join ` K ) |
3 |
|
4that.a |
|- A = ( Atoms ` K ) |
4 |
|
4that.h |
|- H = ( LHyp ` K ) |
5 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
6 |
|
simp21 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
7 |
|
simp22 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
8 |
|
simp32 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> T = ( 0. ` K ) ) |
9 |
|
simp31 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P =/= Q ) |
10 |
|
simp23 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( S e. A /\ -. S .<_ W ) ) |
11 |
|
simp33 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) |
12 |
1 2 3 4
|
4atex2-0aOLDN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ T = ( 0. ` K ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( T .\/ z ) = ( S .\/ z ) ) ) |
13 |
5 6 7 8 9 10 11 12
|
syl133anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( T .\/ z ) = ( S .\/ z ) ) ) |
14 |
|
eqcom |
|- ( ( S .\/ z ) = ( T .\/ z ) <-> ( T .\/ z ) = ( S .\/ z ) ) |
15 |
14
|
anbi2i |
|- ( ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) <-> ( -. z .<_ W /\ ( T .\/ z ) = ( S .\/ z ) ) ) |
16 |
15
|
rexbii |
|- ( E. z e. A ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) <-> E. z e. A ( -. z .<_ W /\ ( T .\/ z ) = ( S .\/ z ) ) ) |
17 |
13 16
|
sylibr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ T = ( 0. ` K ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) ) |