Metamath Proof Explorer

Theorem cdlemg1ltrnlem

Description: Lemma for ltrniotacl . (Contributed by NM, 18-Apr-2013)

Ref Expression
Hypotheses cdlemg1.b 
|- B = ( Base ` K )
cdlemg1.l 
|- .<_ = ( le ` K )
cdlemg1.j 
|- .\/ = ( join ` K )
cdlemg1.m 
|- ./\ = ( meet ` K )
cdlemg1.a 
|- A = ( Atoms ` K )
cdlemg1.h 
|- H = ( LHyp ` K )
cdlemg1.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemg1.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemg1.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemg1.g 
|- G = ( 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 ) )
cdlemg1.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg1.f 
|- F = ( iota_ f e. T ( f ` P ) = Q )
Assertion cdlemg1ltrnlem 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F e. T )

Proof

Step Hyp Ref Expression
1 cdlemg1.b 
 |-  B = ( Base ` K )
2 cdlemg1.l 
 |-  .<_ = ( le ` K )
3 cdlemg1.j 
 |-  .\/ = ( join ` K )
4 cdlemg1.m 
 |-  ./\ = ( meet ` K )
5 cdlemg1.a 
 |-  A = ( Atoms ` K )
6 cdlemg1.h 
 |-  H = ( LHyp ` K )
7 cdlemg1.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdlemg1.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9 cdlemg1.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10 cdlemg1.g 
 |-  G = ( 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 cdlemg1.t 
 |-  T = ( ( LTrn ` K ) ` W )
12 cdlemg1.f 
 |-  F = ( iota_ f e. T ( f ` P ) = Q )
13 1 2 3 4 5 6 7 8 9 10 11 12 cdlemg1b2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F = G )
14 1 2 3 4 5 6 7 8 9 10 11 cdleme50ltrn 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> G e. T )
15 13 14 eqeltrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F e. T )