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Metamath Proof Explorer

Theorem cdlemn5

Description: Part of proof of Lemma N of Crawley p. 121 line 32. (Contributed by NM, 25-Feb-2014)

Ref Expression
Hypotheses cdlemn5.b 
|- B = ( Base ` K )
cdlemn5.l 
|- .<_ = ( le ` K )
cdlemn5.j 
|- .\/ = ( join ` K )
cdlemn5.a 
|- A = ( Atoms ` K )
cdlemn5.h 
|- H = ( LHyp ` K )
cdlemn5.u 
|- U = ( ( DVecH ` K ) ` W )
cdlemn5.s 
|- .(+) = ( LSSum ` U )
cdlemn5.i 
|- I = ( ( DIsoB ` K ) ` W )
cdlemn5.J 
|- J = ( ( DIsoC ` K ) ` W )
Assertion 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 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemn5.b 
 |-  B = ( Base ` K )
2 cdlemn5.l 
 |-  .<_ = ( le ` K )
3 cdlemn5.j 
 |-  .\/ = ( join ` K )
4 cdlemn5.a 
 |-  A = ( Atoms ` K )
5 cdlemn5.h 
 |-  H = ( LHyp ` K )
6 cdlemn5.u 
 |-  U = ( ( DVecH ` K ) ` W )
7 cdlemn5.s 
 |-  .(+) = ( LSSum ` U )
8 cdlemn5.i 
 |-  I = ( ( DIsoB ` K ) ` W )
9 cdlemn5.J 
 |-  J = ( ( DIsoC ` K ) ` W )
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 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
14 eqid 
 |-  ( LSpan ` U ) = ( LSpan ` U )
15 eqid 
 |-  ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = Q ) = ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = Q )
16 eqid 
 |-  ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = R ) = ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = R )
17 eqid 
 |-  ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` Q ) = R ) = ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` Q ) = R )
18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 cdlemn5pre 
 |-  ( ( ( 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 ) ) )