Step |
Hyp |
Ref |
Expression |
1 |
|
4thatlem.ph |
|- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) ) |
2 |
|
4thatlem0.l |
|- .<_ = ( le ` K ) |
3 |
|
4thatlem0.j |
|- .\/ = ( join ` K ) |
4 |
|
4thatlem0.m |
|- ./\ = ( meet ` K ) |
5 |
|
4thatlem0.a |
|- A = ( Atoms ` K ) |
6 |
|
4thatlem0.h |
|- H = ( LHyp ` K ) |
7 |
|
4thatlem0.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
4thatlem0.v |
|- V = ( ( P .\/ S ) ./\ W ) |
9 |
|
4thatlem0.c |
|- C = ( ( Q .\/ T ) ./\ ( P .\/ S ) ) |
10 |
|
4thatlem0.d |
|- D = ( ( R .\/ T ) ./\ ( P .\/ S ) ) |
11 |
1 2 3 5 7
|
4atexlemswapqr |
|- ( ph -> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) ) |
12 |
1 2 3 4 5 6 7 8 9 10
|
4atexlemcnd |
|- ( ph -> C =/= D ) |
13 |
|
pm13.18 |
|- ( ( C = S /\ C =/= D ) -> S =/= D ) |
14 |
13
|
necomd |
|- ( ( C = S /\ C =/= D ) -> D =/= S ) |
15 |
14
|
expcom |
|- ( C =/= D -> ( C = S -> D =/= S ) ) |
16 |
12 15
|
syl |
|- ( ph -> ( C = S -> D =/= S ) ) |
17 |
|
biid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) ) |
18 |
|
eqid |
|- ( ( P .\/ R ) ./\ W ) = ( ( P .\/ R ) ./\ W ) |
19 |
17 2 3 4 5 6 18 8 10
|
4atexlemex2 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) /\ D =/= S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |
20 |
11 16 19
|
syl6an |
|- ( ph -> ( C = S -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) |
21 |
20
|
imp |
|- ( ( ph /\ C = S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) |