Metamath Proof Explorer

Theorem cdlemg1a

Description: Shorter expression for G . TODO: fix comment. TODO: shorten using cdleme or vice-versa? Also, if not shortened with cdleme , then it can be moved up to save repeating hypotheses. (Contributed by NM, 15-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 )
Assertion cdlemg1a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> G = ( iota_ f e. T ( f ` P ) = Q ) )

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 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 )
13 simpll1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) /\ ( f ` P ) = Q ) -> ( K e. HL /\ W e. H ) )
14 simplr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) /\ ( f ` P ) = Q ) -> f e. T )
15 12 ad2antrr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) /\ ( f ` P ) = Q ) -> G e. T )
16 simpll2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) /\ ( f ` P ) = Q ) -> ( P e. A /\ -. P .<_ W ) )
17 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) /\ ( f ` P ) = Q ) -> ( f ` P ) = Q )
18 1 2 3 4 5 6 7 8 9 10 cdleme17d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( G ` P ) = Q )
19 18 ad2antrr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) /\ ( f ` P ) = Q ) -> ( G ` P ) = Q )
20 17 19 eqtr4d 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) /\ ( f ` P ) = Q ) -> ( f ` P ) = ( G ` P ) )
21 2 5 6 11 cdlemd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ f e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( f ` P ) = ( G ` P ) ) -> f = G )
22 13 14 15 16 20 21 syl311anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) /\ ( f ` P ) = Q ) -> f = G )
23 22 ex 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) -> ( ( f ` P ) = Q -> f = G ) )
24 18 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) -> ( G ` P ) = Q )
25 fveq1 
 |-  ( f = G -> ( f ` P ) = ( G ` P ) )
26 25 eqeq1d 
 |-  ( f = G -> ( ( f ` P ) = Q <-> ( G ` P ) = Q ) )
27 24 26 syl5ibrcom 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) -> ( f = G -> ( f ` P ) = Q ) )
28 23 27 impbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ f e. T ) -> ( ( f ` P ) = Q <-> f = G ) )
29 12 28 riota5 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( iota_ f e. T ( f ` P ) = Q ) = G )
30 29 eqcomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> G = ( iota_ f e. T ( f ` P ) = Q ) )