Metamath Proof Explorer

Theorem cdlemeg47rv2

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, 1-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 cdlemeg47rv2 
|- ( ( ( ( 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 ) = ( ( Q .\/ P ) ./\ ( ( G ` S ) .\/ ( ( R .\/ S ) ./\ 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 1 2 3 4 5 6 7 8 9 10 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 )
12 simp22l 
 |-  ( ( ( ( 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 ) ) ) -> R e. A )
13 nfcvd 
 |-  ( R e. A -> F/_ u ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( R .\/ S ) ./\ W ) ) ) )
14 oveq1 
 |-  ( u = R -> ( u .\/ S ) = ( R .\/ S ) )
15 14 oveq1d 
 |-  ( u = R -> ( ( u .\/ S ) ./\ W ) = ( ( R .\/ S ) ./\ W ) )
16 15 oveq2d 
 |-  ( u = R -> ( [_ S / v ]_ N .\/ ( ( u .\/ S ) ./\ W ) ) = ( [_ S / v ]_ N .\/ ( ( R .\/ S ) ./\ W ) ) )
17 16 oveq2d 
 |-  ( u = R -> ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( u .\/ S ) ./\ W ) ) ) = ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( R .\/ S ) ./\ W ) ) ) )
18 13 17 csbiegf 
 |-  ( R e. A -> [_ R / u ]_ ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( u .\/ S ) ./\ W ) ) ) = ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( R .\/ S ) ./\ W ) ) ) )
19 12 18 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 ) ) ) -> [_ R / u ]_ ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( u .\/ S ) ./\ W ) ) ) = ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( R .\/ S ) ./\ W ) ) ) )
20 simp23l 
 |-  ( ( ( ( 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 ) ) ) -> S e. A )
21 eqid 
 |-  ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( u .\/ S ) ./\ W ) ) ) = ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( u .\/ S ) ./\ W ) ) )
22 9 21 cdleme31se2 
 |-  ( S e. A -> [_ S / v ]_ O = ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( u .\/ S ) ./\ W ) ) ) )
23 20 22 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 ) ) ) -> [_ S / v ]_ O = ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( u .\/ S ) ./\ W ) ) ) )
24 23 csbeq2dv 
 |-  ( ( ( ( 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 ) ) ) -> [_ R / u ]_ [_ S / v ]_ O = [_ R / u ]_ ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( u .\/ S ) ./\ W ) ) ) )
25 simp1 
 |-  ( ( ( ( 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 ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
26 simp21 
 |-  ( ( ( ( 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 ) ) ) -> P =/= Q )
27 simp23 
 |-  ( ( ( ( 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 ) ) ) -> ( S e. A /\ -. S .<_ W ) )
28 simp3r 
 |-  ( ( ( ( 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 ) ) ) -> -. S .<_ ( P .\/ Q ) )
29 1 2 3 4 5 6 7 8 9 10 cdlemeg47b 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( G ` S ) = [_ S / v ]_ N )
30 25 26 27 28 29 syl121anc 
 |-  ( ( ( ( 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 ` S ) = [_ S / v ]_ N )
31 30 oveq1d 
 |-  ( ( ( ( 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 ` S ) .\/ ( ( R .\/ S ) ./\ W ) ) = ( [_ S / v ]_ N .\/ ( ( R .\/ S ) ./\ W ) ) )
32 31 oveq2d 
 |-  ( ( ( ( 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 ) ) ) -> ( ( Q .\/ P ) ./\ ( ( G ` S ) .\/ ( ( R .\/ S ) ./\ W ) ) ) = ( ( Q .\/ P ) ./\ ( [_ S / v ]_ N .\/ ( ( R .\/ S ) ./\ W ) ) ) )
33 19 24 32 3eqtr4d 
 |-  ( ( ( ( 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 ) ) ) -> [_ R / u ]_ [_ S / v ]_ O = ( ( Q .\/ P ) ./\ ( ( G ` S ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )
34 11 33 eqtrd 
 |-  ( ( ( ( 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 ) = ( ( Q .\/ P ) ./\ ( ( G ` S ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )