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Metamath Proof Explorer

Theorem cdlemg2cex

Description: Any translation is one of our F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf ? (Contributed by NM, 22-Apr-2013)

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 )
cdlemg2ex.u 
|- U = ( ( p .\/ q ) ./\ W )
cdlemg2ex.d 
|- D = ( ( t .\/ U ) ./\ ( q .\/ ( ( p .\/ t ) ./\ W ) ) )
cdlemg2ex.e 
|- E = ( ( p .\/ q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
cdlemg2ex.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 cdlemg2cex 
|- ( ( K e. HL /\ W e. H ) -> ( F e. T <-> E. p e. A E. q e. A ( -. p .<_ W /\ -. q .<_ W /\ 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 cdlemg2ex.u 
 |-  U = ( ( p .\/ q ) ./\ W )
9 cdlemg2ex.d 
 |-  D = ( ( t .\/ U ) ./\ ( q .\/ ( ( p .\/ t ) ./\ W ) ) )
10 cdlemg2ex.e 
 |-  E = ( ( p .\/ q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
11 cdlemg2ex.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 cdlemg1cex 
 |-  ( ( K e. HL /\ W e. H ) -> ( F e. T <-> E. p e. A E. q e. A ( -. p .<_ W /\ -. q .<_ W /\ F = ( iota_ f e. T ( f ` p ) = q ) ) ) )
13 simplll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ q e. A ) ) /\ ( -. p .<_ W /\ -. q .<_ W ) ) -> K e. HL )
14 simpllr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ q e. A ) ) /\ ( -. p .<_ W /\ -. q .<_ W ) ) -> W e. H )
15 simplrl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ q e. A ) ) /\ ( -. p .<_ W /\ -. q .<_ W ) ) -> p e. A )
16 simprl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ q e. A ) ) /\ ( -. p .<_ W /\ -. q .<_ W ) ) -> -. p .<_ W )
17 simplrr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ q e. A ) ) /\ ( -. p .<_ W /\ -. q .<_ W ) ) -> q e. A )
18 simprr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ q e. A ) ) /\ ( -. p .<_ W /\ -. q .<_ W ) ) -> -. q .<_ W )
19 eqid 
 |-  ( iota_ f e. T ( f ` p ) = q ) = ( iota_ f e. T ( f ` p ) = q )
20 1 2 3 4 5 6 8 9 10 11 7 19 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 )
21 13 14 15 16 17 18 20 syl222anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ q e. A ) ) /\ ( -. p .<_ W /\ -. q .<_ W ) ) -> ( iota_ f e. T ( f ` p ) = q ) = G )
22 21 eqeq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ q e. A ) ) /\ ( -. p .<_ W /\ -. q .<_ W ) ) -> ( F = ( iota_ f e. T ( f ` p ) = q ) <-> F = G ) )
23 22 pm5.32da 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ q e. A ) ) -> ( ( ( -. p .<_ W /\ -. q .<_ W ) /\ F = ( iota_ f e. T ( f ` p ) = q ) ) <-> ( ( -. p .<_ W /\ -. q .<_ W ) /\ F = G ) ) )
24 df-3an 
 |-  ( ( -. p .<_ W /\ -. q .<_ W /\ F = ( iota_ f e. T ( f ` p ) = q ) ) <-> ( ( -. p .<_ W /\ -. q .<_ W ) /\ F = ( iota_ f e. T ( f ` p ) = q ) ) )
25 df-3an 
 |-  ( ( -. p .<_ W /\ -. q .<_ W /\ F = G ) <-> ( ( -. p .<_ W /\ -. q .<_ W ) /\ F = G ) )
26 23 24 25 3bitr4g 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( p e. A /\ q e. A ) ) -> ( ( -. p .<_ W /\ -. q .<_ W /\ F = ( iota_ f e. T ( f ` p ) = q ) ) <-> ( -. p .<_ W /\ -. q .<_ W /\ F = G ) ) )
27 26 2rexbidva 
 |-  ( ( K e. HL /\ W e. H ) -> ( E. p e. A E. q e. A ( -. p .<_ W /\ -. q .<_ W /\ F = ( iota_ f e. T ( f ` p ) = q ) ) <-> E. p e. A E. q e. A ( -. p .<_ W /\ -. q .<_ W /\ F = G ) ) )
28 12 27 bitrd 
 |-  ( ( K e. HL /\ W e. H ) -> ( F e. T <-> E. p e. A E. q e. A ( -. p .<_ W /\ -. q .<_ W /\ F = G ) ) )