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Metamath Proof Explorer

Theorem dicelval2nd

Description: Membership in value of the partial isomorphism C for a lattice K . (Contributed by NM, 16-Feb-2014)

Ref Expression
Hypotheses dicelval2nd.l 
|- .<_ = ( le ` K )
dicelval2nd.a 
|- A = ( Atoms ` K )
dicelval2nd.h 
|- H = ( LHyp ` K )
dicelval2nd.e 
|- E = ( ( TEndo ` K ) ` W )
dicelval2nd.i 
|- I = ( ( DIsoC ` K ) ` W )
Assertion dicelval2nd 
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> ( 2nd ` Y ) e. E )

Proof

Step Hyp Ref Expression
1 dicelval2nd.l 
 |-  .<_ = ( le ` K )
2 dicelval2nd.a 
 |-  A = ( Atoms ` K )
3 dicelval2nd.h 
 |-  H = ( LHyp ` K )
4 dicelval2nd.e 
 |-  E = ( ( TEndo ` K ) ` W )
5 dicelval2nd.i 
 |-  I = ( ( DIsoC ` K ) ` W )
6 eqid 
 |-  ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
7 eqid 
 |-  ( Base ` ( ( DVecH ` K ) ` W ) ) = ( Base ` ( ( DVecH ` K ) ` W ) )
8 1 2 3 5 6 7 dicssdvh 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( Base ` ( ( DVecH ` K ) ` W ) ) )
9 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
10 3 9 4 6 7 dvhvbase 
 |-  ( ( K e. HL /\ W e. H ) -> ( Base ` ( ( DVecH ` K ) ` W ) ) = ( ( ( LTrn ` K ) ` W ) X. E ) )
11 10 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Base ` ( ( DVecH ` K ) ` W ) ) = ( ( ( LTrn ` K ) ` W ) X. E ) )
12 8 11 sseqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) C_ ( ( ( LTrn ` K ) ` W ) X. E ) )
13 12 sseld 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) -> Y e. ( ( ( LTrn ` K ) ` W ) X. E ) ) )
14 13 3impia 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> Y e. ( ( ( LTrn ` K ) ` W ) X. E ) )
15 xp2nd 
 |-  ( Y e. ( ( ( LTrn ` K ) ` W ) X. E ) -> ( 2nd ` Y ) e. E )
16 14 15 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> ( 2nd ` Y ) e. E )