Metamath Proof Explorer

Theorem cdlemefs29clN

Description: Show closure of the unique element in cdleme29c . (Contributed by NM, 27-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemefs32.b 
|- B = ( Base ` K )
cdlemefs32.l 
|- .<_ = ( le ` K )
cdlemefs32.j 
|- .\/ = ( join ` K )
cdlemefs32.m 
|- ./\ = ( meet ` K )
cdlemefs32.a 
|- A = ( Atoms ` K )
cdlemefs32.h 
|- H = ( LHyp ` K )
cdlemefs32.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemefs32.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs32.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemefs32.i 
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
cdlemefs32.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , C )
cdlemefs29cl.o 
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) -> z = ( N .\/ ( R ./\ W ) ) ) )
Assertion cdlemefs29clN 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> O e. B )

Proof

Step Hyp Ref Expression
1 cdlemefs32.b 
 |-  B = ( Base ` K )
2 cdlemefs32.l 
 |-  .<_ = ( le ` K )
3 cdlemefs32.j 
 |-  .\/ = ( join ` K )
4 cdlemefs32.m 
 |-  ./\ = ( meet ` K )
5 cdlemefs32.a 
 |-  A = ( Atoms ` K )
6 cdlemefs32.h 
 |-  H = ( LHyp ` K )
7 cdlemefs32.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemefs32.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemefs32.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10 cdlemefs32.i 
 |-  I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
11 cdlemefs32.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , C )
12 cdlemefs29cl.o 
 |-  O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) -> z = ( N .\/ ( R ./\ W ) ) ) )
13 breq1 
 |-  ( s = R -> ( s .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ Q ) ) )
14 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
15 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> s e. A )
16 simp3rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> -. s .<_ W )
17 15 16 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> ( s e. A /\ -. s .<_ W ) )
18 simp3rr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> s .<_ ( P .\/ Q ) )
19 simp2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> P =/= Q )
20 1 2 3 4 5 6 7 8 9 10 11 cdlemefs27cl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )
21 14 17 18 19 20 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q /\ ( s e. A /\ ( -. s .<_ W /\ s .<_ ( P .\/ Q ) ) ) ) -> N e. B )
22 1 2 3 4 5 6 7 8 9 10 11 cdlemefs32snb 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N e. B )
23 1 2 3 4 5 6 13 21 22 12 cdlemefrs29clN 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> O e. B )