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Metamath Proof Explorer

Theorem cdlemefr29clN

Description: Show closure of the unique element in cdleme29c . TODO fix comment. TODO Not needed? (Contributed by NM, 29-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemefr27.b 
|- B = ( Base ` K )
cdlemefr27.l 
|- .<_ = ( le ` K )
cdlemefr27.j 
|- .\/ = ( join ` K )
cdlemefr27.m 
|- ./\ = ( meet ` K )
cdlemefr27.a 
|- A = ( Atoms ` K )
cdlemefr27.h 
|- H = ( LHyp ` K )
cdlemefr27.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemefr27.c 
|- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdlemefr27.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , C )
cdlemefr29cl.o 
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) -> z = ( N .\/ ( R ./\ W ) ) ) )
Assertion cdlemefr29clN 
|- ( ( ( ( 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 cdlemefr27.b 
 |-  B = ( Base ` K )
2 cdlemefr27.l 
 |-  .<_ = ( le ` K )
3 cdlemefr27.j 
 |-  .\/ = ( join ` K )
4 cdlemefr27.m 
 |-  ./\ = ( meet ` K )
5 cdlemefr27.a 
 |-  A = ( Atoms ` K )
6 cdlemefr27.h 
 |-  H = ( LHyp ` K )
7 cdlemefr27.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemefr27.c 
 |-  C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdlemefr27.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , C )
10 cdlemefr29cl.o 
 |-  O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( R ./\ W ) ) = R ) -> z = ( N .\/ ( R ./\ W ) ) ) )
11 breq1 
 |-  ( s = R -> ( s .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ Q ) ) )
12 11 notbid 
 |-  ( s = R -> ( -. s .<_ ( P .\/ Q ) <-> -. R .<_ ( P .\/ Q ) ) )
13 simp11 
 |-  ( ( ( ( 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 ) )
14 simp12l 
 |-  ( ( ( ( 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 e. A )
15 simp13l 
 |-  ( ( ( ( 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 ) ) ) ) -> Q e. A )
16 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 )
17 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 ) )
18 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 )
19 1 2 3 4 5 6 7 8 9 cdlemefr27cl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( s e. A /\ -. s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )
20 13 14 15 16 17 18 19 syl33anc 
 |-  ( ( ( ( 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 )
21 1 2 3 4 5 6 7 8 9 cdlemefr32snb 
 |-  ( ( ( ( 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 )
22 1 2 3 4 5 6 12 20 21 10 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 )