Step |
Hyp |
Ref |
Expression |
1 |
|
4at.l |
|- .<_ = ( le ` K ) |
2 |
|
4at.j |
|- .\/ = ( join ` K ) |
3 |
|
4at.a |
|- A = ( Atoms ` K ) |
4 |
|
simprr |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) |
5 |
|
simprl |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) |
6 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) ) |
7 |
|
simpl21 |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> R e. A ) |
8 |
|
simpl23 |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> V e. A ) |
9 |
|
simpl31 |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> W e. A ) |
10 |
|
simpl32 |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> -. R .<_ ( ( P .\/ Q ) .\/ W ) ) |
11 |
1 2 3
|
4atlem10a |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ V e. A /\ W e. A ) /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) ) -> ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ W ) ) = ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) |
12 |
6 7 8 9 10 11
|
syl131anc |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ W ) ) = ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) |
13 |
5 12
|
mpbid |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ W ) ) = ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) |
14 |
4 13
|
breqtrrd |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> S .<_ ( ( P .\/ Q ) .\/ ( R .\/ W ) ) ) |
15 |
|
simpl22 |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> S e. A ) |
16 |
|
simpl33 |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> -. S .<_ ( ( P .\/ Q ) .\/ R ) ) |
17 |
1 2 3
|
4atlem9 |
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ W e. A ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) -> ( S .<_ ( ( P .\/ Q ) .\/ ( R .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ Q ) .\/ ( R .\/ W ) ) ) ) |
18 |
6 7 15 9 16 17
|
syl131anc |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> ( S .<_ ( ( P .\/ Q ) .\/ ( R .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ Q ) .\/ ( R .\/ W ) ) ) ) |
19 |
14 18
|
mpbid |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ Q ) .\/ ( R .\/ W ) ) ) |
20 |
19 13
|
eqtrd |
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ V e. A ) /\ ( W e. A /\ -. R .<_ ( ( P .\/ Q ) .\/ W ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) /\ ( R .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) |