Metamath Proof Explorer

Theorem cdleme11dN

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme11 . (Contributed by NM, 13-Jun-2012) (New usage is discouraged.)

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 cdleme11dN 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P .\/ S ) =/= ( P .\/ 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 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) )
8 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) )
9 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> -. S .<_ ( P .\/ Q ) )
10 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> U .<_ ( S .\/ T ) )
11 1 2 3 4 5 6 cdleme11c 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> -. P .<_ ( S .\/ T ) )
12 7 8 9 10 11 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> -. P .<_ ( S .\/ T ) )
13 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> K e. HL )
14 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> P e. A )
15 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> S e. A )
16 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ S e. A ) -> S .<_ ( P .\/ S ) )
17 13 14 15 16 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> S .<_ ( P .\/ S ) )
18 breq2 
 |-  ( ( P .\/ S ) = ( P .\/ T ) -> ( S .<_ ( P .\/ S ) <-> S .<_ ( P .\/ T ) ) )
19 17 18 syl5ibcom 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( P .\/ S ) = ( P .\/ T ) -> S .<_ ( P .\/ T ) ) )
20 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> T e. A )
21 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> S =/= T )
22 1 2 4 hlatexch2 
 |-  ( ( K e. HL /\ ( S e. A /\ P e. A /\ T e. A ) /\ S =/= T ) -> ( S .<_ ( P .\/ T ) -> P .<_ ( S .\/ T ) ) )
23 13 15 14 20 21 22 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( S .<_ ( P .\/ T ) -> P .<_ ( S .\/ T ) ) )
24 19 23 syld 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( P .\/ S ) = ( P .\/ T ) -> P .<_ ( S .\/ T ) ) )
25 24 necon3bd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( -. P .<_ ( S .\/ T ) -> ( P .\/ S ) =/= ( P .\/ T ) ) )
26 12 25 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P .\/ S ) =/= ( P .\/ T ) )