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 |
1 2 3 4 5 6 7 8
|
4atexlemtlw |
|- ( ph -> T .<_ W ) |
10 |
1
|
4atexlemkc |
|- ( ph -> K e. CvLat ) |
11 |
1 2 3 4 5 6 7
|
4atexlemu |
|- ( ph -> U e. A ) |
12 |
1 2 3 4 5 6 7 8
|
4atexlemv |
|- ( ph -> V e. A ) |
13 |
1
|
4atexlemt |
|- ( ph -> T e. A ) |
14 |
1 2 3 4 5 6 7 8
|
4atexlemunv |
|- ( ph -> U =/= V ) |
15 |
1
|
4atexlemutvt |
|- ( ph -> ( U .\/ T ) = ( V .\/ T ) ) |
16 |
5 3
|
cvlsupr5 |
|- ( ( K e. CvLat /\ ( U e. A /\ V e. A /\ T e. A ) /\ ( U =/= V /\ ( U .\/ T ) = ( V .\/ T ) ) ) -> T =/= U ) |
17 |
10 11 12 13 14 15 16
|
syl132anc |
|- ( ph -> T =/= U ) |
18 |
17
|
adantr |
|- ( ( ph /\ T .<_ ( P .\/ Q ) ) -> T =/= U ) |
19 |
1
|
4atexlemk |
|- ( ph -> K e. HL ) |
20 |
1
|
4atexlemw |
|- ( ph -> W e. H ) |
21 |
19 20
|
jca |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
22 |
21
|
adantr |
|- ( ( ph /\ T .<_ ( P .\/ Q ) ) -> ( K e. HL /\ W e. H ) ) |
23 |
1
|
4atexlempw |
|- ( ph -> ( P e. A /\ -. P .<_ W ) ) |
24 |
23
|
adantr |
|- ( ( ph /\ T .<_ ( P .\/ Q ) ) -> ( P e. A /\ -. P .<_ W ) ) |
25 |
1
|
4atexlemq |
|- ( ph -> Q e. A ) |
26 |
25
|
adantr |
|- ( ( ph /\ T .<_ ( P .\/ Q ) ) -> Q e. A ) |
27 |
13
|
adantr |
|- ( ( ph /\ T .<_ ( P .\/ Q ) ) -> T e. A ) |
28 |
1
|
4atexlempnq |
|- ( ph -> P =/= Q ) |
29 |
28
|
adantr |
|- ( ( ph /\ T .<_ ( P .\/ Q ) ) -> P =/= Q ) |
30 |
|
simpr |
|- ( ( ph /\ T .<_ ( P .\/ Q ) ) -> T .<_ ( P .\/ Q ) ) |
31 |
2 3 4 5 6 7
|
lhpat3 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( Q e. A /\ T e. A ) /\ ( P =/= Q /\ T .<_ ( P .\/ Q ) ) ) -> ( -. T .<_ W <-> T =/= U ) ) |
32 |
22 24 26 27 29 30 31
|
syl222anc |
|- ( ( ph /\ T .<_ ( P .\/ Q ) ) -> ( -. T .<_ W <-> T =/= U ) ) |
33 |
18 32
|
mpbird |
|- ( ( ph /\ T .<_ ( P .\/ Q ) ) -> -. T .<_ W ) |
34 |
33
|
ex |
|- ( ph -> ( T .<_ ( P .\/ Q ) -> -. T .<_ W ) ) |
35 |
9 34
|
mt2d |
|- ( ph -> -. T .<_ ( P .\/ Q ) ) |