Metamath Proof Explorer

Theorem cdlemeg47rv

Description: Value of g_s(r) when r is an atom under pq and s is any atom not under pq, using very compact hypotheses. TODO: FIX COMMENT. (Contributed by NM, 3-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 cdlemeg47rv 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( G ` R ) = [_ R / u ]_ [_ S / v ]_ O )

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 3 5 cdleme46f2g1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( Q =/= P /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( Q .\/ P ) /\ -. S .<_ ( Q .\/ P ) ) ) )
12 1 2 3 4 5 6 7 8 10 9 cdlemefs45 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( Q =/= P /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( Q .\/ P ) /\ -. S .<_ ( Q .\/ P ) ) ) -> ( G ` R ) = [_ R / u ]_ [_ S / v ]_ O )
13 11 12 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( G ` R ) = [_ R / u ]_ [_ S / v ]_ O )