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Metamath Proof Explorer

Theorem cdlemn

Description: Lemma N of Crawley p. 121 line 27. (Contributed by NM, 27-Feb-2014)

Ref Expression
Hypotheses cdlemn11.b 
|- B = ( Base ` K )
cdlemn11.l 
|- .<_ = ( le ` K )
cdlemn11.j 
|- .\/ = ( join ` K )
cdlemn11.a 
|- A = ( Atoms ` K )
cdlemn11.h 
|- H = ( LHyp ` K )
cdlemn11.i 
|- I = ( ( DIsoB ` K ) ` W )
cdlemn11.J 
|- J = ( ( DIsoC ` K ) ` W )
cdlemn11.u 
|- U = ( ( DVecH ` K ) ` W )
cdlemn11.s 
|- .(+) = ( LSSum ` U )
Assertion cdlemn 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) ) -> ( R .<_ ( Q .\/ X ) <-> ( J ` R ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemn11.b 
 |-  B = ( Base ` K )
2 cdlemn11.l 
 |-  .<_ = ( le ` K )
3 cdlemn11.j 
 |-  .\/ = ( join ` K )
4 cdlemn11.a 
 |-  A = ( Atoms ` K )
5 cdlemn11.h 
 |-  H = ( LHyp ` K )
6 cdlemn11.i 
 |-  I = ( ( DIsoB ` K ) ` W )
7 cdlemn11.J 
 |-  J = ( ( DIsoC ` K ) ` W )
8 cdlemn11.u 
 |-  U = ( ( DVecH ` K ) ` W )
9 cdlemn11.s 
 |-  .(+) = ( LSSum ` U )
10 1 2 3 4 5 8 9 6 7 cdlemn5 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( J ` R ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) )
11 10 3expia 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) ) -> ( R .<_ ( Q .\/ X ) -> ( J ` R ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) )
12 1 2 3 4 5 6 7 8 9 cdlemn11 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ ( J ` R ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) -> R .<_ ( Q .\/ X ) )
13 12 3expia 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) ) -> ( ( J ` R ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) -> R .<_ ( Q .\/ X ) ) )
14 11 13 impbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) ) -> ( R .<_ ( Q .\/ X ) <-> ( J ` R ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) )