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Metamath Proof Explorer

Theorem cdlemefr32snb

Description: Show closure of [_ R / s ]_ N . (Contributed by NM, 28-Mar-2013)

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 )
Assertion 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 )

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 1 2 3 4 5 6 7 8 9 cdlemefr32sn2aw 
 |-  ( ( ( ( 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. A /\ -. [_ R / s ]_ N .<_ W ) )
11 10 simpld 
 |-  ( ( ( ( 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. A )
12 1 5 atbase 
 |-  ( [_ R / s ]_ N e. A -> [_ R / s ]_ N e. B )
13 11 12 syl 
 |-  ( ( ( ( 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 )