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Metamath Proof Explorer

Theorem 4atexlemntlpq

Description: Lemma for 4atexlem7 . (Contributed by NM, 24-Nov-2012)

Ref Expression
Hypotheses 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 ) ) ) )
4thatlem0.l 
|- .<_ = ( le ` K )
4thatlem0.j 
|- .\/ = ( join ` K )
4thatlem0.m 
|- ./\ = ( meet ` K )
4thatlem0.a 
|- A = ( Atoms ` K )
4thatlem0.h 
|- H = ( LHyp ` K )
4thatlem0.u 
|- U = ( ( P .\/ Q ) ./\ W )
4thatlem0.v 
|- V = ( ( P .\/ S ) ./\ W )
Assertion 4atexlemntlpq 
|- ( ph -> -. T .<_ ( P .\/ Q ) )

Proof

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 ) )