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Metamath Proof Explorer

Theorem cdleme11a

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme11 . (Contributed by NM, 12-Jun-2012)

Ref Expression
Hypotheses cdleme11.l 
|- .<_ = ( le ` K )
cdleme11.j 
|- .\/ = ( join ` K )
cdleme11.m 
|- ./\ = ( meet ` K )
cdleme11.a 
|- A = ( Atoms ` K )
cdleme11.h 
|- H = ( LHyp ` K )
cdleme11.u 
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdleme11a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> ( S .\/ U ) = ( S .\/ T ) )

Proof

Step Hyp Ref Expression
1 cdleme11.l 
 |-  .<_ = ( le ` K )
2 cdleme11.j 
 |-  .\/ = ( join ` K )
3 cdleme11.m 
 |-  ./\ = ( meet ` K )
4 cdleme11.a 
 |-  A = ( Atoms ` K )
5 cdleme11.h 
 |-  H = ( LHyp ` K )
6 cdleme11.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 simp3rr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> U .<_ ( S .\/ T ) )
8 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> K e. HL )
9 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> ( K e. HL /\ W e. H ) )
10 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> ( Q e. A /\ P =/= Q ) )
12 1 2 3 4 5 6 lhpat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
13 9 10 11 12 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> U e. A )
14 simp3rl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> T e. A )
15 simp3ll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> S e. A )
16 simp2ll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> P e. A )
17 simp2rl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> Q e. A )
18 simp3l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> ( S e. A /\ -. S .<_ W ) )
19 1 2 3 4 5 6 cdleme0c 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ W ) ) -> U =/= S )
20 9 16 17 18 19 syl121anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> U =/= S )
21 1 2 4 hlatexchb1 
 |-  ( ( K e. HL /\ ( U e. A /\ T e. A /\ S e. A ) /\ U =/= S ) -> ( U .<_ ( S .\/ T ) <-> ( S .\/ U ) = ( S .\/ T ) ) )
22 8 13 14 15 20 21 syl131anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> ( U .<_ ( S .\/ T ) <-> ( S .\/ U ) = ( S .\/ T ) ) )
23 7 22 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> ( S .\/ U ) = ( S .\/ T ) )