Metamath Proof Explorer

Theorem cdlemn11

Description: Part of proof of Lemma N of Crawley p. 121 line 37. (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 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 ) )

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 eqid 
 |-  ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
11 eqid 
 |-  ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
12 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
13 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
14 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
15 eqid 
 |-  ( +g ` U ) = ( +g ` U )
16 eqid 
 |-  ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = Q ) = ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = Q )
17 eqid 
 |-  ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = R ) = ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = R )
18 1 2 3 4 5 10 11 12 13 14 6 7 8 15 9 16 17 cdlemn11pre 
 |-  ( ( ( 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 ) )