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Metamath Proof Explorer

Theorem cdleme21d

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 115, 3rd line. D , F , N , E , B , Z represent s_2, f(s), f_s(r), z_2, f(z), f_z(r) respectively. We prove f_s(r) = f_z(r). (Contributed by NM, 29-Nov-2012)

Ref Expression
Hypotheses cdleme21.l 
|- .<_ = ( le ` K )
cdleme21.j 
|- .\/ = ( join ` K )
cdleme21.m 
|- ./\ = ( meet ` K )
cdleme21.a 
|- A = ( Atoms ` K )
cdleme21.h 
|- H = ( LHyp ` K )
cdleme21.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme21.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme21.b 
|- B = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
cdleme21.d 
|- D = ( ( R .\/ S ) ./\ W )
cdleme21.e 
|- E = ( ( R .\/ z ) ./\ W )
cdleme21d.n 
|- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
cdleme21d.z 
|- Z = ( ( P .\/ Q ) ./\ ( B .\/ E ) )
Assertion cdleme21d 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> N = Z )

Proof

Step Hyp Ref Expression
1 cdleme21.l 
 |-  .<_ = ( le ` K )
2 cdleme21.j 
 |-  .\/ = ( join ` K )
3 cdleme21.m 
 |-  ./\ = ( meet ` K )
4 cdleme21.a 
 |-  A = ( Atoms ` K )
5 cdleme21.h 
 |-  H = ( LHyp ` K )
6 cdleme21.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme21.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme21.b 
 |-  B = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
9 cdleme21.d 
 |-  D = ( ( R .\/ S ) ./\ W )
10 cdleme21.e 
 |-  E = ( ( R .\/ z ) ./\ W )
11 cdleme21d.n 
 |-  N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
12 cdleme21d.z 
 |-  Z = ( ( P .\/ Q ) ./\ ( B .\/ E ) )
13 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> ( K e. HL /\ W e. H ) )
14 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
15 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
16 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> ( R e. A /\ -. R .<_ W ) )
17 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> ( S e. A /\ -. S .<_ W ) )
18 simp33l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> ( z e. A /\ -. z .<_ W ) )
19 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> P =/= Q )
20 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> K e. HL )
21 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> P e. A )
22 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> Q e. A )
23 simp2rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> S e. A )
24 simp32l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> -. S .<_ ( P .\/ Q ) )
25 18 simpld 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> z e. A )
26 simp33r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> ( P .\/ z ) = ( S .\/ z ) )
27 1 2 4 cdleme21a 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> S =/= z )
28 20 21 22 23 24 25 26 27 syl322anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> S =/= z )
29 19 28 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> ( P =/= Q /\ S =/= z ) )
30 1 2 4 cdleme21b 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> -. z .<_ ( P .\/ Q ) )
31 20 21 22 23 19 24 25 26 30 syl332anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> -. z .<_ ( P .\/ Q ) )
32 simp32r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> R .<_ ( P .\/ Q ) )
33 24 31 32 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> ( -. S .<_ ( P .\/ Q ) /\ -. z .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) )
34 1 2 3 4 5 6 cdleme21c 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> -. U .<_ ( S .\/ z ) )
35 13 14 22 23 19 24 25 26 34 syl332anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> -. U .<_ ( S .\/ z ) )
36 eqid 
 |-  ( ( S .\/ z ) ./\ W ) = ( ( S .\/ z ) ./\ W )
37 1 2 3 4 5 6 7 8 9 10 36 11 12 cdleme20 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( z e. A /\ -. z .<_ W ) ) /\ ( ( P =/= Q /\ S =/= z ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. z .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ -. U .<_ ( S .\/ z ) ) ) -> N = Z )
38 13 14 15 16 17 18 29 33 35 37 syl333anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> N = Z )