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Metamath Proof Explorer

Theorem dih1dimc

Description: Isomorphism H at an atom not under W . (Contributed by NM, 27-Apr-2014)

Ref Expression
Hypotheses dih1dimc.l 
|- .<_ = ( le ` K )
dih1dimc.a 
|- A = ( Atoms ` K )
dih1dimc.h 
|- H = ( LHyp ` K )
dih1dimc.p 
|- P = ( ( oc ` K ) ` W )
dih1dimc.t 
|- T = ( ( LTrn ` K ) ` W )
dih1dimc.i 
|- I = ( ( DIsoH ` K ) ` W )
dih1dimc.u 
|- U = ( ( DVecH ` K ) ` W )
dih1dimc.n 
|- N = ( LSpan ` U )
dih1dimc.f 
|- F = ( iota_ f e. T ( f ` P ) = Q )
Assertion dih1dimc 
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = ( N ` { <. F , ( _I |` T ) >. } ) )

Proof

Step Hyp Ref Expression
1 dih1dimc.l 
 |-  .<_ = ( le ` K )
2 dih1dimc.a 
 |-  A = ( Atoms ` K )
3 dih1dimc.h 
 |-  H = ( LHyp ` K )
4 dih1dimc.p 
 |-  P = ( ( oc ` K ) ` W )
5 dih1dimc.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 dih1dimc.i 
 |-  I = ( ( DIsoH ` K ) ` W )
7 dih1dimc.u 
 |-  U = ( ( DVecH ` K ) ` W )
8 dih1dimc.n 
 |-  N = ( LSpan ` U )
9 dih1dimc.f 
 |-  F = ( iota_ f e. T ( f ` P ) = Q )
10 eqid 
 |-  ( ( DIsoC ` K ) ` W ) = ( ( DIsoC ` K ) ` W )
11 1 2 3 10 6 dihvalcqat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = ( ( ( DIsoC ` K ) ` W ) ` Q ) )
12 1 2 3 4 5 10 7 8 9 diclspsn 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( ( DIsoC ` K ) ` W ) ` Q ) = ( N ` { <. F , ( _I |` T ) >. } ) )
13 11 12 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = ( N ` { <. F , ( _I |` T ) >. } ) )