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Metamath Proof Explorer

Theorem cdleme46frvlpq

Description: Show that ( FS ) is not under P .\/ Q when S isn't. (Contributed by NM, 1-Apr-2013)

Ref Expression
Hypotheses cdlemef46.b 
|- B = ( Base ` K )
cdlemef46.l 
|- .<_ = ( le ` K )
cdlemef46.j 
|- .\/ = ( join ` K )
cdlemef46.m 
|- ./\ = ( meet ` K )
cdlemef46.a 
|- A = ( Atoms ` K )
cdlemef46.h 
|- H = ( LHyp ` K )
cdlemef46.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemef46.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemefs46.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemef46.f 
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
Assertion cdleme46frvlpq 
|- ( ( ( ( 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 ) ) -> -. ( F ` S ) .<_ ( P .\/ Q ) )

Proof

Step Hyp Ref Expression
1 cdlemef46.b 
 |-  B = ( Base ` K )
2 cdlemef46.l 
 |-  .<_ = ( le ` K )
3 cdlemef46.j 
 |-  .\/ = ( join ` K )
4 cdlemef46.m 
 |-  ./\ = ( meet ` K )
5 cdlemef46.a 
 |-  A = ( Atoms ` K )
6 cdlemef46.h 
 |-  H = ( LHyp ` K )
7 cdlemef46.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemef46.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemefs46.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10 cdlemef46.f 
 |-  F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
11 eqid 
 |-  ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
12 2 3 4 5 6 7 11 cdleme35fnpq 
 |-  ( ( ( ( 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 .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .<_ ( P .\/ Q ) )
13 1 2 3 4 5 6 7 8 10 cdlemefr45e 
 |-  ( ( ( ( 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 ) ) -> ( F ` S ) = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
14 13 breq1d 
 |-  ( ( ( ( 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 ) ) -> ( ( F ` S ) .<_ ( P .\/ Q ) <-> ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .<_ ( P .\/ Q ) ) )
15 12 14 mtbird 
 |-  ( ( ( ( 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 ) ) -> -. ( F ` S ) .<_ ( P .\/ Q ) )