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Metamath Proof Explorer

Theorem cdleme42g

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 8-Mar-2013)

Ref Expression
Hypotheses cdleme41.b 
|- B = ( Base ` K )
cdleme41.l 
|- .<_ = ( le ` K )
cdleme41.j 
|- .\/ = ( join ` K )
cdleme41.m 
|- ./\ = ( meet ` K )
cdleme41.a 
|- A = ( Atoms ` K )
cdleme41.h 
|- H = ( LHyp ` K )
cdleme41.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme41.d 
|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
cdleme41.e 
|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme41.g 
|- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
cdleme41.i 
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
cdleme41.n 
|- N = if ( s .<_ ( P .\/ Q ) , I , D )
cdleme41.o 
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
cdleme41.f 
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
cdleme34e.v 
|- V = ( ( R .\/ S ) ./\ W )
Assertion cdleme42g 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( F ` ( R .\/ S ) ) = ( ( F ` R ) .\/ V ) )

Proof

Step Hyp Ref Expression
1 cdleme41.b 
 |-  B = ( Base ` K )
2 cdleme41.l 
 |-  .<_ = ( le ` K )
3 cdleme41.j 
 |-  .\/ = ( join ` K )
4 cdleme41.m 
 |-  ./\ = ( meet ` K )
5 cdleme41.a 
 |-  A = ( Atoms ` K )
6 cdleme41.h 
 |-  H = ( LHyp ` K )
7 cdleme41.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 cdleme41.d 
 |-  D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9 cdleme41.e 
 |-  E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
10 cdleme41.g 
 |-  G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) )
11 cdleme41.i 
 |-  I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) )
12 cdleme41.n 
 |-  N = if ( s .<_ ( P .\/ Q ) , I , D )
13 cdleme41.o 
 |-  O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
14 cdleme41.f 
 |-  F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
15 cdleme34e.v 
 |-  V = ( ( R .\/ S ) ./\ W )
16 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
17 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( R e. A /\ -. R .<_ W ) )
18 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( S e. A /\ -. S .<_ W ) )
19 1 2 3 4 5 6 15 cdleme42a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) -> ( R .\/ S ) = ( R .\/ V ) )
20 16 17 18 19 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( R .\/ S ) = ( R .\/ V ) )
21 20 fveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( F ` ( R .\/ S ) ) = ( F ` ( R .\/ V ) ) )
22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 cdleme42f 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( F ` ( R .\/ V ) ) = ( ( F ` R ) .\/ V ) )
23 21 22 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ P =/= Q ) -> ( F ` ( R .\/ S ) ) = ( ( F ` R ) .\/ V ) )