Metamath Proof Explorer

Theorem cdleme43cN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT p. 115 last line: r v g(s) = r v v_2. (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 cdleme43cN 
|- ( ( ( ( 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 ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( R .\/ D ) = ( R .\/ Y ) )

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 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 ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( K e. HL /\ W e. H ) )
17 simp22 
 |-  ( ( ( ( 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 ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( R e. A /\ -. R .<_ W ) )
18 simp1 
 |-  ( ( ( ( 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 ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
19 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 ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P =/= Q )
20 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 ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( S e. A /\ -. S .<_ W ) )
21 simp3 
 |-  ( ( ( ( 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 ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. S .<_ ( P .\/ Q ) )
22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 cdleme43bN 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( D e. A /\ -. D .<_ W ) )
23 18 19 20 21 22 syl121anc 
 |-  ( ( ( ( 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 ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( D e. A /\ -. D .<_ W ) )
24 1 2 3 4 5 6 15 cdleme42a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( D e. A /\ -. D .<_ W ) ) -> ( R .\/ D ) = ( R .\/ Y ) )
25 16 17 23 24 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 ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( R .\/ D ) = ( R .\/ Y ) )