Step |
Hyp |
Ref |
Expression |
1 |
|
islpln2a.l |
|- .<_ = ( le ` K ) |
2 |
|
islpln2a.j |
|- .\/ = ( join ` K ) |
3 |
|
islpln2a.a |
|- A = ( Atoms ` K ) |
4 |
|
islpln2a.p |
|- P = ( LPlanes ` K ) |
5 |
|
islpln2a.y |
|- Y = ( ( Q .\/ R ) .\/ S ) |
6 |
1 2 3
|
3noncolr1N |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( S =/= Q /\ -. R .<_ ( S .\/ Q ) ) ) |
7 |
6
|
simprd |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> -. R .<_ ( S .\/ Q ) ) |
8 |
7
|
3expia |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) -> -. R .<_ ( S .\/ Q ) ) ) |
9 |
1 2 3 4 5
|
islpln2ah |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Y e. P <-> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) ) |
10 |
2 3
|
hlatjcom |
|- ( ( K e. HL /\ Q e. A /\ S e. A ) -> ( Q .\/ S ) = ( S .\/ Q ) ) |
11 |
10
|
3adant3r2 |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Q .\/ S ) = ( S .\/ Q ) ) |
12 |
11
|
breq2d |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( R .<_ ( Q .\/ S ) <-> R .<_ ( S .\/ Q ) ) ) |
13 |
12
|
notbid |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( -. R .<_ ( Q .\/ S ) <-> -. R .<_ ( S .\/ Q ) ) ) |
14 |
8 9 13
|
3imtr4d |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Y e. P -> -. R .<_ ( Q .\/ S ) ) ) |
15 |
14
|
3impia |
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> -. R .<_ ( Q .\/ S ) ) |