Metamath Proof Explorer


Theorem dihmeetlem14N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem14.b
|- B = ( Base ` K )
dihmeetlem14.l
|- .<_ = ( le ` K )
dihmeetlem14.h
|- H = ( LHyp ` K )
dihmeetlem14.j
|- .\/ = ( join ` K )
dihmeetlem14.m
|- ./\ = ( meet ` K )
dihmeetlem14.a
|- A = ( Atoms ` K )
dihmeetlem14.u
|- U = ( ( DVecH ` K ) ` W )
dihmeetlem14.s
|- .(+) = ( LSSum ` U )
dihmeetlem14.i
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihmeetlem14N
|- ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ p e. B ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( ( I ` ( Y ./\ p ) ) .(+) ( ( I ` r ) i^i ( I ` p ) ) ) = ( ( I ` Y ) i^i ( I ` p ) ) )

Proof

Step Hyp Ref Expression
1 dihmeetlem14.b
 |-  B = ( Base ` K )
2 dihmeetlem14.l
 |-  .<_ = ( le ` K )
3 dihmeetlem14.h
 |-  H = ( LHyp ` K )
4 dihmeetlem14.j
 |-  .\/ = ( join ` K )
5 dihmeetlem14.m
 |-  ./\ = ( meet ` K )
6 dihmeetlem14.a
 |-  A = ( Atoms ` K )
7 dihmeetlem14.u
 |-  U = ( ( DVecH ` K ) ` W )
8 dihmeetlem14.s
 |-  .(+) = ( LSSum ` U )
9 dihmeetlem14.i
 |-  I = ( ( DIsoH ` K ) ` W )
10 1 2 3 4 5 6 7 8 9 dihmeetlem12N
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ p e. B ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( ( I ` ( Y ./\ p ) ) .(+) ( ( I ` r ) i^i ( I ` p ) ) ) = ( ( I ` Y ) i^i ( I ` p ) ) )