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Metamath Proof Explorer

Theorem cdlemefs27cl

Description: Part of proof of Lemma E in Crawley p. 113. Closure of N . TODO FIX COMMENT This is the start of a re-proof of cdleme27cl etc. with the s .<_ ( P .\/ Q ) condition (so as to not have the C hypothesis). (Contributed by NM, 24-Mar-2013)

Ref Expression
Hypotheses cdlemefs26.b 
|- B = ( Base ` K )
cdlemefs26.l 
|- .<_ = ( le ` K )
cdlemefs26.j 
|- .\/ = ( join ` K )
cdlemefs26.m 
|- ./\ = ( meet ` K )
cdlemefs26.a 
|- A = ( Atoms ` K )
cdlemefs26.h 
|- H = ( LHyp ` K )
cdlemefs27.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemefs27.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs27.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemefs27.i 
|- I = ( iota_ u e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> u = E ) )
cdlemefs27.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , C )
Assertion cdlemefs27cl 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )

Proof

Step Hyp Ref Expression
1 cdlemefs26.b 
 |-  B = ( Base ` K )
2 cdlemefs26.l 
 |-  .<_ = ( le ` K )
3 cdlemefs26.j 
 |-  .\/ = ( join ` K )
4 cdlemefs26.m 
 |-  ./\ = ( meet ` K )
5 cdlemefs26.a 
 |-  A = ( Atoms ` K )
6 cdlemefs26.h 
 |-  H = ( LHyp ` K )
7 cdlemefs27.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemefs27.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemefs27.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10 cdlemefs27.i 
 |-  I = ( iota_ u e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> u = E ) )
11 cdlemefs27.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , C )
12 simpr2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> s .<_ ( P .\/ Q ) )
13 12 iftrued 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> if ( s .<_ ( P .\/ Q ) , I , C ) = I )
14 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> ( K e. HL /\ W e. H ) )
15 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> ( P e. A /\ -. P .<_ W ) )
16 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> ( Q e. A /\ -. Q .<_ W ) )
17 simpr1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> ( s e. A /\ -. s .<_ W ) )
18 simpr3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> P =/= Q )
19 1 2 3 4 5 6 7 8 9 10 cdleme25cl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( s e. A /\ -. s .<_ W ) /\ ( P =/= Q /\ s .<_ ( P .\/ Q ) ) ) -> I e. B )
20 14 15 16 17 18 12 19 syl312anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> I e. B )
21 13 20 eqeltrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> if ( s .<_ ( P .\/ Q ) , I , C ) e. B )
22 11 21 eqeltrid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )