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Metamath Proof Explorer

Theorem cdlemg2dN

Description: This theorem can be used to shorten G = hypothesis. TODO: Fix comment. (Contributed by NM, 21-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 cdlemg2dN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> 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 simp3l 
 |-  ( ( ( 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 )
13 simp1 
 |-  ( ( ( 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 simp2l 
 |-  ( ( ( 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 ) )
15 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> ( Q e. A /\ -. Q .<_ W ) )
16 simp3r 
 |-  ( ( ( 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 )
17 1 2 3 4 5 6 7 8 9 10 11 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 ) )
18 13 14 15 16 17 syl31anc 
 |-  ( ( ( 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 <-> F = G ) )
19 12 18 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ ( F ` P ) = Q ) ) -> F = G )