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Metamath Proof Explorer

Theorem cdleme43dN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT p. 116 2nd line: f(r) v s = f(r) v f(g(s)). (Contributed by NM, 20-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme43.b 
|- B = ( Base ` K )
cdleme43.l 
|- .<_ = ( le ` K )
cdleme43.j 
|- .\/ = ( join ` K )
cdleme43.m 
|- ./\ = ( meet ` K )
cdleme43.a 
|- A = ( Atoms ` K )
cdleme43.h 
|- H = ( LHyp ` K )
cdleme43.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme43.x 
|- X = ( ( Q .\/ P ) ./\ W )
cdleme43.c 
|- C = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme43.f 
|- Z = ( ( P .\/ Q ) ./\ ( C .\/ ( ( R .\/ S ) ./\ W ) ) )
cdleme43.d 
|- D = ( ( S .\/ X ) ./\ ( P .\/ ( ( Q .\/ S ) ./\ W ) ) )
cdleme43.g 
|- G = ( ( Q .\/ P ) ./\ ( D .\/ ( ( Z .\/ S ) ./\ W ) ) )
cdleme43.e 
|- E = ( ( D .\/ U ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) )
cdleme43.v 
|- V = ( ( Z .\/ S ) ./\ W )
cdleme43.y 
|- Y = ( ( R .\/ D ) ./\ W )
Assertion cdleme43dN 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( Z .\/ S ) = ( Z .\/ E ) )

Proof

Step Hyp Ref Expression
1 cdleme43.b 
 |-  B = ( Base ` K )
2 cdleme43.l 
 |-  .<_ = ( le ` K )
3 cdleme43.j 
 |-  .\/ = ( join ` K )
4 cdleme43.m 
 |-  ./\ = ( meet ` K )
5 cdleme43.a 
 |-  A = ( Atoms ` K )
6 cdleme43.h 
 |-  H = ( LHyp ` K )
7 cdleme43.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme43.x 
 |-  X = ( ( Q .\/ P ) ./\ W )
9 cdleme43.c 
 |-  C = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
10 cdleme43.f 
 |-  Z = ( ( P .\/ Q ) ./\ ( C .\/ ( ( R .\/ S ) ./\ W ) ) )
11 cdleme43.d 
 |-  D = ( ( S .\/ X ) ./\ ( P .\/ ( ( Q .\/ S ) ./\ W ) ) )
12 cdleme43.g 
 |-  G = ( ( Q .\/ P ) ./\ ( D .\/ ( ( Z .\/ S ) ./\ W ) ) )
13 cdleme43.e 
 |-  E = ( ( D .\/ U ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) )
14 cdleme43.v 
 |-  V = ( ( Z .\/ S ) ./\ W )
15 cdleme43.y 
 |-  Y = ( ( R .\/ D ) ./\ W )
16 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> K e. HL )
17 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P e. A )
18 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> Q e. A )
19 3 5 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
20 16 17 18 19 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
21 20 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ W ) = ( ( Q .\/ P ) ./\ W ) )
22 21 7 8 3eqtr4g 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> U = X )
23 22 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( D .\/ U ) = ( D .\/ X ) )
24 23 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( D .\/ U ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) ) = ( ( D .\/ X ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) ) )
25 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
26 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
27 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
28 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> P =/= Q )
29 28 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> Q =/= P )
30 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( S e. A /\ -. S .<_ W ) )
31 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( P .\/ Q ) )
32 20 breq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( S .<_ ( P .\/ Q ) <-> S .<_ ( Q .\/ P ) ) )
33 31 32 mtbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> -. S .<_ ( Q .\/ P ) )
34 2 3 4 5 6 8 11 cdleme35g 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( Q =/= P /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( Q .\/ P ) ) -> ( ( D .\/ X ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) ) = S )
35 25 26 27 29 30 33 34 syl321anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( D .\/ X ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) ) = S )
36 24 35 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( D .\/ U ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) ) = S )
37 13 36 eqtrid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> E = S )
38 37 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( Z .\/ E ) = ( Z .\/ S ) )
39 38 eqcomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( Z .\/ S ) = ( Z .\/ E ) )