This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdleme25cl

Description: Show closure of the unique element in cdleme25c . (Contributed by NM, 2-Feb-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 ) ) )
cdleme25cl.i 
|- I = ( iota_ u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = N ) )
Assertion cdleme25cl 
|- ( ( ( ( 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 ) ) ) -> I e. B )

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 cdleme25cl.i 
 |-  I = ( iota_ u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = N ) )
11 1 2 3 4 5 6 7 8 9 cdleme25c 
 |-  ( ( ( ( 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 ) )
12 riotacl 
 |-  ( E! u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = N ) -> ( iota_ u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = N ) ) e. B )
13 11 12 syl 
 |-  ( ( ( ( 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 ) ) ) -> ( iota_ u e. B A. s e. A ( ( -. s .<_ W /\ -. s .<_ ( P .\/ Q ) ) -> u = N ) ) e. B )
14 10 13 eqeltrid 
 |-  ( ( ( ( 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 ) ) ) -> I e. B )