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Metamath Proof Explorer

Theorem cdlemednuN

Description: Part of proof of Lemma E in Crawley p. 113. Utility lemma. D represents s_2. (Contributed by NM, 18-Nov-2012) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 cdlemeda.l 
 |-  .<_ = ( le ` K )
2 cdlemeda.j 
 |-  .\/ = ( join ` K )
3 cdlemeda.m 
 |-  ./\ = ( meet ` K )
4 cdlemeda.a 
 |-  A = ( Atoms ` K )
5 cdlemeda.h 
 |-  H = ( LHyp ` K )
6 cdlemeda.d 
 |-  D = ( ( R .\/ S ) ./\ W )
7 cdlemednu.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 1 2 3 4 5 6 cdlemednpq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. D .<_ ( P .\/ Q ) )
9 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
10 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> W e. H )
11 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A )
12 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. A )
13 1 2 3 4 5 7 cdlemeulpq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) )
14 9 10 11 12 13 syl22anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> U .<_ ( P .\/ Q ) )
15 breq1 
 |-  ( D = U -> ( D .<_ ( P .\/ Q ) <-> U .<_ ( P .\/ Q ) ) )
16 14 15 syl5ibrcom 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( D = U -> D .<_ ( P .\/ Q ) ) )
17 16 necon3bd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( -. D .<_ ( P .\/ Q ) -> D =/= U ) )
18 8 17 mpd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> D =/= U )