Metamath Proof Explorer

Theorem cdlemefr27cl

Description: Part of proof of Lemma E in Crawley p. 113. Closure of N . (Contributed by NM, 23-Mar-2013)

Ref Expression
Hypotheses cdlemefr27.b 
|- B = ( Base ` K )
cdlemefr27.l 
|- .<_ = ( le ` K )
cdlemefr27.j 
|- .\/ = ( join ` K )
cdlemefr27.m 
|- ./\ = ( meet ` K )
cdlemefr27.a 
|- A = ( Atoms ` K )
cdlemefr27.h 
|- H = ( LHyp ` K )
cdlemefr27.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemefr27.c 
|- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdlemefr27.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , C )
Assertion cdlemefr27cl 
|- ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )

Proof

Step Hyp Ref Expression
1 cdlemefr27.b 
 |-  B = ( Base ` K )
2 cdlemefr27.l 
 |-  .<_ = ( le ` K )
3 cdlemefr27.j 
 |-  .\/ = ( join ` K )
4 cdlemefr27.m 
 |-  ./\ = ( meet ` K )
5 cdlemefr27.a 
 |-  A = ( Atoms ` K )
6 cdlemefr27.h 
 |-  H = ( LHyp ` K )
7 cdlemefr27.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemefr27.c 
 |-  C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdlemefr27.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , C )
10 simpr2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> -. s .<_ ( P .\/ Q ) )
11 10 iffalsed 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> if ( s .<_ ( P .\/ Q ) , I , C ) = C )
12 9 11 syl5eq 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N = C )
13 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> K e. HL )
14 simpl1r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> W e. H )
15 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> P e. A )
16 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> Q e. A )
17 simpr1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> s e. A )
18 2 3 4 5 6 7 8 1 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ s e. A ) ) -> C e. B )
19 13 14 15 16 17 18 syl23anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> C e. B )
20 12 19 eqeltrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )