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Metamath Proof Explorer

Theorem cdleme4gfv

Description: Part of proof of Lemma D in Crawley p. 113. TODO: Can this replace uses of cdleme32a ? TODO: Can this be used to help prove the R or S case where X is an atom? TODO: Would an antecedent transformer like cdleme46f2g2 help? (Contributed by NM, 8-Apr-2013)

Ref Expression
Hypotheses cdlemef47.b 
|- B = ( Base ` K )
cdlemef47.l 
|- .<_ = ( le ` K )
cdlemef47.j 
|- .\/ = ( join ` K )
cdlemef47.m 
|- ./\ = ( meet ` K )
cdlemef47.a 
|- A = ( Atoms ` K )
cdlemef47.h 
|- H = ( LHyp ` K )
cdlemef47.v 
|- V = ( ( Q .\/ P ) ./\ W )
cdlemef47.n 
|- N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) )
cdlemefs47.o 
|- O = ( ( Q .\/ P ) ./\ ( N .\/ ( ( u .\/ v ) ./\ W ) ) )
cdlemef47.g 
|- G = ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = O ) ) , [_ u / v ]_ N ) .\/ ( a ./\ W ) ) ) ) , a ) )
Assertion cdleme4gfv 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( G ` X ) = ( ( G ` S ) .\/ ( X ./\ W ) ) )

Proof

Step Hyp Ref Expression
1 cdlemef47.b 
 |-  B = ( Base ` K )
2 cdlemef47.l 
 |-  .<_ = ( le ` K )
3 cdlemef47.j 
 |-  .\/ = ( join ` K )
4 cdlemef47.m 
 |-  ./\ = ( meet ` K )
5 cdlemef47.a 
 |-  A = ( Atoms ` K )
6 cdlemef47.h 
 |-  H = ( LHyp ` K )
7 cdlemef47.v 
 |-  V = ( ( Q .\/ P ) ./\ W )
8 cdlemef47.n 
 |-  N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) )
9 cdlemefs47.o 
 |-  O = ( ( Q .\/ P ) ./\ ( N .\/ ( ( u .\/ v ) ./\ W ) ) )
10 cdlemef47.g 
 |-  G = ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = O ) ) , [_ u / v ]_ N ) .\/ ( a ./\ W ) ) ) ) , a ) )
11 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( K e. HL /\ W e. H ) )
12 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( Q e. A /\ -. Q .<_ W ) )
13 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( P e. A /\ -. P .<_ W ) )
14 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> P =/= Q )
15 14 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> Q =/= P )
16 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( X e. B /\ -. X .<_ W ) )
17 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) )
18 1 2 3 4 5 6 7 8 9 10 cdleme48fv 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( Q =/= P /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( G ` X ) = ( ( G ` S ) .\/ ( X ./\ W ) ) )
19 11 12 13 15 16 17 18 syl321anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( X e. B /\ -. X .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( S .\/ ( X ./\ W ) ) = X ) ) -> ( G ` X ) = ( ( G ` S ) .\/ ( X ./\ W ) ) )