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Metamath Proof Explorer

Theorem cdleme43fsv1sn

Description: Value of [_ R / s ]_ N when R .<_ ( P .\/ Q ) . (Contributed by NM, 30-Mar-2013)

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 )
cdleme43fs.y 
|- Y = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme43fs.z 
|- Z = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( R .\/ S ) ./\ W ) ) )
Assertion cdleme43fsv1sn 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ N = Z )

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 cdleme43fs.y 
 |-  Y = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
13 cdleme43fs.z 
 |-  Z = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( R .\/ S ) ./\ W ) ) )
14 eqid 
 |-  ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) )
15 eqid 
 |-  ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) ) ) ) = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = ( ( P .\/ Q ) ./\ ( D .\/ ( ( R .\/ t ) ./\ W ) ) ) ) )
16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 cdleme43fsv1snlem 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ N = Z )