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Metamath Proof Explorer

Theorem cdleme25b

Description: Transform cdleme24 . TODO get rid of $d's on U , N (Contributed by NM, 1-Jan-2013)

Ref Expression
Hypotheses cdleme24.b 
|- B = ( Base ` K )
cdleme24.l 
|- .<_ = ( le ` K )
cdleme24.j 
|- .\/ = ( join ` K )
cdleme24.m 
|- ./\ = ( meet ` K )
cdleme24.a 
|- A = ( Atoms ` K )
cdleme24.h 
|- H = ( LHyp ` K )
cdleme24.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme24.f 
|- F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme24.n 
|- N = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ s ) ./\ W ) ) )
Assertion cdleme25b 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) ) ) -> E. u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = N ) )

Proof

Step Hyp Ref Expression
1 cdleme24.b 
 |-  B = ( Base ` K )
2 cdleme24.l 
 |-  .<_ = ( le ` K )
3 cdleme24.j 
 |-  .\/ = ( join ` K )
4 cdleme24.m 
 |-  ./\ = ( meet ` K )
5 cdleme24.a 
 |-  A = ( Atoms ` K )
6 cdleme24.h 
 |-  H = ( LHyp ` K )
7 cdleme24.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme24.f 
 |-  F = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme24.n 
 |-  N = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ s ) ./\ W ) ) )
10 1 2 3 4 5 6 7 8 9 cdleme25a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) ) ) -> E. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) /\ N e. B ) )
11 eqid 
 |-  ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
12 eqid 
 |-  ( ( P .\/ Q ) ./\ ( ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) .\/ ( ( R .\/ t ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) .\/ ( ( R .\/ t ) ./\ W ) ) )
13 1 2 3 4 5 6 7 8 9 11 12 cdleme24 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) ) ) -> A. s e. A A. t e. A ( ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) /\ ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) ) -> N = ( ( P .\/ Q ) ./\ ( ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) .\/ ( ( R .\/ t ) ./\ W ) ) ) ) )
14 breq1 
 |-  ( s = t -> ( s .<_ W <-> t .<_ W ) )
15 14 notbid 
 |-  ( s = t -> ( -. s .<_ W <-> -. t .<_ W ) )
16 breq1 
 |-  ( s = t -> ( s .<_ ( P .\/ Q ) <-> t .<_ ( P .\/ Q ) ) )
17 16 notbid 
 |-  ( s = t -> ( -. s .<_ ( P .\/ Q ) <-> -. t .<_ ( P .\/ Q ) ) )
18 15 17 anbi12d 
 |-  ( s = t -> ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) <-> ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) ) )
19 oveq1 
 |-  ( s = t -> ( s .\/ U ) = ( t .\/ U ) )
20 oveq2 
 |-  ( s = t -> ( P .\/ s ) = ( P .\/ t ) )
21 20 oveq1d 
 |-  ( s = t -> ( ( P .\/ s ) ./\ W ) = ( ( P .\/ t ) ./\ W ) )
22 21 oveq2d 
 |-  ( s = t -> ( Q .\/ ( ( P .\/ s ) ./\ W ) ) = ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
23 19 22 oveq12d 
 |-  ( s = t -> ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) )
24 8 23 eqtrid 
 |-  ( s = t -> F = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) )
25 oveq2 
 |-  ( s = t -> ( R .\/ s ) = ( R .\/ t ) )
26 25 oveq1d 
 |-  ( s = t -> ( ( R .\/ s ) ./\ W ) = ( ( R .\/ t ) ./\ W ) )
27 24 26 oveq12d 
 |-  ( s = t -> ( F .\/ ( ( R .\/ s ) ./\ W ) ) = ( ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) .\/ ( ( R .\/ t ) ./\ W ) ) )
28 27 oveq2d 
 |-  ( s = t -> ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ s ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) .\/ ( ( R .\/ t ) ./\ W ) ) ) )
29 9 28 eqtrid 
 |-  ( s = t -> N = ( ( P .\/ Q ) ./\ ( ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) .\/ ( ( R .\/ t ) ./\ W ) ) ) )
30 18 29 reusv3 
 |-  ( E. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) /\ N e. B ) -> ( A. s e. A A. t e. A ( ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) /\ ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) ) -> N = ( ( P .\/ Q ) ./\ ( ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) .\/ ( ( R .\/ t ) ./\ W ) ) ) ) <-> E. u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = N ) ) )
31 30 biimpd 
 |-  ( E. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) /\ N e. B ) -> ( A. s e. A A. t e. A ( ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) /\ ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) ) -> N = ( ( P .\/ Q ) ./\ ( ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) .\/ ( ( R .\/ t ) ./\ W ) ) ) ) -> E. u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = N ) ) )
32 10 13 31 sylc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) /\ ( P =/= Q /\ R .<_ ( P .\/ Q ) ) ) -> E. u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = N ) )