Metamath Proof Explorer

Theorem diameetN

Description: Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses diam.m 
|- ./\ = ( meet ` K )
diam.h 
|- H = ( LHyp ` K )
diam.i 
|- I = ( ( DIsoA ` K ) ` W )
Assertion diameetN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )

Proof

Step Hyp Ref Expression
1 diam.m 
 |-  ./\ = ( meet ` K )
2 diam.h 
 |-  H = ( LHyp ` K )
3 diam.i 
 |-  I = ( ( DIsoA ` K ) ` W )
4 eqid 
 |-  ( glb ` K ) = ( glb ` K )
5 simpll 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> K e. HL )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 2 3 diadmclN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> X e. ( Base ` K ) )
8 7 adantrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> X e. ( Base ` K ) )
9 6 2 3 diadmclN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. dom I ) -> Y e. ( Base ` K ) )
10 9 adantrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> Y e. ( Base ` K ) )
11 4 1 5 8 10 meetval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( X ./\ Y ) = ( ( glb ` K ) ` { X , Y } ) )
12 11 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( X ./\ Y ) ) = ( I ` ( ( glb ` K ) ` { X , Y } ) ) )
13 simpl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( K e. HL /\ W e. H ) )
14 prssi 
 |-  ( ( X e. dom I /\ Y e. dom I ) -> { X , Y } C_ dom I )
15 14 adantl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> { X , Y } C_ dom I )
16 prnzg 
 |-  ( X e. dom I -> { X , Y } =/= (/) )
17 16 ad2antrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> { X , Y } =/= (/) )
18 4 2 3 diaglbN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( { X , Y } C_ dom I /\ { X , Y } =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( I ` x ) )
19 13 15 17 18 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( I ` x ) )
20 fveq2 
 |-  ( x = X -> ( I ` x ) = ( I ` X ) )
21 fveq2 
 |-  ( x = Y -> ( I ` x ) = ( I ` Y ) )
22 20 21 iinxprg 
 |-  ( ( X e. dom I /\ Y e. dom I ) -> |^|_ x e. { X , Y } ( I ` x ) = ( ( I ` X ) i^i ( I ` Y ) ) )
23 22 adantl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> |^|_ x e. { X , Y } ( I ` x ) = ( ( I ` X ) i^i ( I ` Y ) ) )
24 12 19 23 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )