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Metamath Proof Explorer

Theorem dihvalc

Description: Value of isomorphism H for a lattice K when -. X .<_ W . (Contributed by NM, 4-Mar-2014)

Ref Expression
Hypotheses dihval.b 
|- B = ( Base ` K )
dihval.l 
|- .<_ = ( le ` K )
dihval.j 
|- .\/ = ( join ` K )
dihval.m 
|- ./\ = ( meet ` K )
dihval.a 
|- A = ( Atoms ` K )
dihval.h 
|- H = ( LHyp ` K )
dihval.i 
|- I = ( ( DIsoH ` K ) ` W )
dihval.d 
|- D = ( ( DIsoB ` K ) ` W )
dihval.c 
|- C = ( ( DIsoC ` K ) ` W )
dihval.u 
|- U = ( ( DVecH ` K ) ` W )
dihval.s 
|- S = ( LSubSp ` U )
dihval.p 
|- .(+) = ( LSSum ` U )
Assertion dihvalc 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( I ` X ) = ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 dihval.b 
 |-  B = ( Base ` K )
2 dihval.l 
 |-  .<_ = ( le ` K )
3 dihval.j 
 |-  .\/ = ( join ` K )
4 dihval.m 
 |-  ./\ = ( meet ` K )
5 dihval.a 
 |-  A = ( Atoms ` K )
6 dihval.h 
 |-  H = ( LHyp ` K )
7 dihval.i 
 |-  I = ( ( DIsoH ` K ) ` W )
8 dihval.d 
 |-  D = ( ( DIsoB ` K ) ` W )
9 dihval.c 
 |-  C = ( ( DIsoC ` K ) ` W )
10 dihval.u 
 |-  U = ( ( DVecH ` K ) ` W )
11 dihval.s 
 |-  S = ( LSubSp ` U )
12 dihval.p 
 |-  .(+) = ( LSSum ` U )
13 1 2 3 4 5 6 7 8 9 10 11 12 dihval 
 |-  ( ( ( K e. V /\ W e. H ) /\ X e. B ) -> ( I ` X ) = if ( X .<_ W , ( D ` X ) , ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) ) )
14 iffalse 
 |-  ( -. X .<_ W -> if ( X .<_ W , ( D ` X ) , ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) ) = ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) )
15 13 14 sylan9eq 
 |-  ( ( ( ( K e. V /\ W e. H ) /\ X e. B ) /\ -. X .<_ W ) -> ( I ` X ) = ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) )
16 15 anasss 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( I ` X ) = ( iota_ u e. S A. q e. A ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> u = ( ( C ` q ) .(+) ( D ` ( X ./\ W ) ) ) ) ) )