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Metamath Proof Explorer

Theorem cdleme21k

Description: Eliminate S =/= T condition in cdleme21 . (Contributed by NM, 26-Dec-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 ) ) )
cdleme21g.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme21g.d 
|- D = ( ( R .\/ S ) ./\ W )
cdleme21g.y 
|- Y = ( ( R .\/ T ) ./\ W )
cdleme21g.n 
|- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
cdleme21g.o 
|- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
Assertion cdleme21k 
|- ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> N = O )

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 cdleme21g.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 cdleme21g.d 
 |-  D = ( ( R .\/ S ) ./\ W )
10 cdleme21g.y 
 |-  Y = ( ( R .\/ T ) ./\ W )
11 cdleme21g.n 
 |-  N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
12 cdleme21g.o 
 |-  O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
13 oveq1 
 |-  ( S = T -> ( S .\/ U ) = ( T .\/ U ) )
14 oveq2 
 |-  ( S = T -> ( P .\/ S ) = ( P .\/ T ) )
15 14 oveq1d 
 |-  ( S = T -> ( ( P .\/ S ) ./\ W ) = ( ( P .\/ T ) ./\ W ) )
16 15 oveq2d 
 |-  ( S = T -> ( Q .\/ ( ( P .\/ S ) ./\ W ) ) = ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
17 13 16 oveq12d 
 |-  ( S = T -> ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) ) )
18 17 7 8 3eqtr4g 
 |-  ( S = T -> F = G )
19 oveq2 
 |-  ( S = T -> ( R .\/ S ) = ( R .\/ T ) )
20 19 oveq1d 
 |-  ( S = T -> ( ( R .\/ S ) ./\ W ) = ( ( R .\/ T ) ./\ W ) )
21 20 9 10 3eqtr4g 
 |-  ( S = T -> D = Y )
22 18 21 oveq12d 
 |-  ( S = T -> ( F .\/ D ) = ( G .\/ Y ) )
23 22 oveq2d 
 |-  ( S = T -> ( ( P .\/ Q ) ./\ ( F .\/ D ) ) = ( ( P .\/ Q ) ./\ ( G .\/ Y ) ) )
24 23 11 12 3eqtr4g 
 |-  ( S = T -> N = O )
25 24 eqeq1d 
 |-  ( S = T -> ( N = O <-> O = O ) )
26 simpl11 
 |-  ( ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( K e. HL /\ W e. H ) )
27 simpl12 
 |-  ( ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( P e. A /\ -. P .<_ W ) )
28 simpl13 
 |-  ( ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( Q e. A /\ -. Q .<_ W ) )
29 simpl21 
 |-  ( ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( R e. A /\ -. R .<_ W ) )
30 simpl22 
 |-  ( ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( S e. A /\ -. S .<_ W ) )
31 simpl23 
 |-  ( ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( T e. A /\ -. T .<_ W ) )
32 simpl3l 
 |-  ( ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> P =/= Q )
33 simpr 
 |-  ( ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> S =/= T )
34 32 33 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( P =/= Q /\ S =/= T ) )
35 simpl3r 
 |-  ( ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 cdleme21 
 |-  ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> N = O )
37 26 27 28 29 30 31 34 35 36 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> N = O )
38 eqidd 
 |-  ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> O = O )
39 25 37 38 pm2.61ne 
 |-  ( ( ( ( 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 ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> N = O )