Metamath Proof Explorer


Theorem cdleme19e

Description: Part of proof of Lemma E in Crawley p. 113, 5th paragraph on p. 114, line 2. D , F , Y , G represent s_2, f(s), t_2, f(t). We prove f(s) \/ s_2=f(t) \/ t_2. (Contributed by NM, 14-Nov-2012)

Ref Expression
Hypotheses cdleme19.l
|- .<_ = ( le ` K )
cdleme19.j
|- .\/ = ( join ` K )
cdleme19.m
|- ./\ = ( meet ` K )
cdleme19.a
|- A = ( Atoms ` K )
cdleme19.h
|- H = ( LHyp ` K )
cdleme19.u
|- U = ( ( P .\/ Q ) ./\ W )
cdleme19.f
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme19.g
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme19.d
|- D = ( ( R .\/ S ) ./\ W )
cdleme19.y
|- Y = ( ( R .\/ T ) ./\ W )
Assertion cdleme19e
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( F .\/ D ) = ( G .\/ Y ) )

Proof

Step Hyp Ref Expression
1 cdleme19.l
 |-  .<_ = ( le ` K )
2 cdleme19.j
 |-  .\/ = ( join ` K )
3 cdleme19.m
 |-  ./\ = ( meet ` K )
4 cdleme19.a
 |-  A = ( Atoms ` K )
5 cdleme19.h
 |-  H = ( LHyp ` K )
6 cdleme19.u
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme19.f
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme19.g
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 cdleme19.d
 |-  D = ( ( R .\/ S ) ./\ W )
10 cdleme19.y
 |-  Y = ( ( R .\/ T ) ./\ W )
11 simp11l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> K e. HL )
12 11 hllatd
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> K e. Lat )
13 simp11r
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> W e. H )
14 simp12l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> P e. A )
15 simp13l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> Q e. A )
16 simp21l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> S e. A )
17 eqid
 |-  ( Base ` K ) = ( Base ` K )
18 1 2 3 4 5 6 7 17 cdleme1b
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> F e. ( Base ` K ) )
19 11 13 14 15 16 18 syl23anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> F e. ( Base ` K ) )
20 simp22l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> T e. A )
21 1 2 3 4 5 6 8 17 cdleme1b
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ T e. A ) ) -> G e. ( Base ` K ) )
22 11 13 14 15 20 21 syl23anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> G e. ( Base ` K ) )
23 17 2 latjcom
 |-  ( ( K e. Lat /\ F e. ( Base ` K ) /\ G e. ( Base ` K ) ) -> ( F .\/ G ) = ( G .\/ F ) )
24 12 19 22 23 syl3anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( F .\/ G ) = ( G .\/ F ) )
25 1 2 3 4 5 6 7 8 9 10 cdleme19d
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( F .\/ D ) = ( F .\/ G ) )
26 simp11
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( K e. HL /\ W e. H ) )
27 simp12
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
28 simp13
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
29 simp22
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( T e. A /\ -. T .<_ W ) )
30 simp21
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( S e. A /\ -. S .<_ W ) )
31 simp23
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> R e. A )
32 simp31l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> P =/= Q )
33 simp31r
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> S =/= T )
34 33 necomd
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> T =/= S )
35 32 34 jca
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( P =/= Q /\ T =/= S ) )
36 simp32r
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> -. T .<_ ( P .\/ Q ) )
37 simp32l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> -. S .<_ ( P .\/ Q ) )
38 36 37 jca
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) )
39 simp33l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> R .<_ ( P .\/ Q ) )
40 simp33r
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> R .<_ ( S .\/ T ) )
41 2 4 hlatjcom
 |-  ( ( K e. HL /\ S e. A /\ T e. A ) -> ( S .\/ T ) = ( T .\/ S ) )
42 11 16 20 41 syl3anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( S .\/ T ) = ( T .\/ S ) )
43 40 42 breqtrd
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> R .<_ ( T .\/ S ) )
44 39 43 jca
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( R .<_ ( P .\/ Q ) /\ R .<_ ( T .\/ S ) ) )
45 1 2 3 4 5 6 8 7 10 9 cdleme19d
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( T e. A /\ -. T .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ T =/= S ) /\ ( -. T .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( T .\/ S ) ) ) ) -> ( G .\/ Y ) = ( G .\/ F ) )
46 26 27 28 29 30 31 35 38 44 45 syl333anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( G .\/ Y ) = ( G .\/ F ) )
47 24 25 46 3eqtr4d
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( F .\/ D ) = ( G .\/ Y ) )