Metamath Proof Explorer

Theorem cdlemg2cN

Description: Any translation belongs to the set of functions constructed for cdleme . TODO: Fix comment. (Contributed by NM, 22-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg2.b 
|- B = ( Base ` K )
cdlemg2.l 
|- .<_ = ( le ` K )
cdlemg2.j 
|- .\/ = ( join ` K )
cdlemg2.m 
|- ./\ = ( meet ` K )
cdlemg2.a 
|- A = ( Atoms ` K )
cdlemg2.h 
|- H = ( LHyp ` K )
cdlemg2.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg2.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdlemg2.d 
|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdlemg2.e 
|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemg2.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 ) )
Assertion cdlemg2cN 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) -> ( F e. T <-> F = G ) )

Proof

Step Hyp Ref Expression
1 cdlemg2.b 
 |-  B = ( Base ` K )
2 cdlemg2.l 
 |-  .<_ = ( le ` K )
3 cdlemg2.j 
 |-  .\/ = ( join ` K )
4 cdlemg2.m 
 |-  ./\ = ( meet ` K )
5 cdlemg2.a 
 |-  A = ( Atoms ` K )
6 cdlemg2.h 
 |-  H = ( LHyp ` K )
7 cdlemg2.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemg2.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
9 cdlemg2.d 
 |-  D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
10 cdlemg2.e 
 |-  E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
11 cdlemg2.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 ) )
12 2 5 6 7 cdlemg1cN 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) -> ( F e. T <-> F = ( iota_ f e. T ( f ` P ) = Q ) ) )
13 eqid 
 |-  ( iota_ f e. T ( f ` P ) = Q ) = ( iota_ f e. T ( f ` P ) = Q )
14 1 2 3 4 5 6 8 9 10 11 7 13 cdlemg1b2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( iota_ f e. T ( f ` P ) = Q ) = G )
15 14 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) -> ( iota_ f e. T ( f ` P ) = Q ) = G )
16 15 eqeq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) -> ( F = ( iota_ f e. T ( f ` P ) = Q ) <-> F = G ) )
17 12 16 bitrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) -> ( F e. T <-> F = G ) )