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Metamath Proof Explorer

Theorem dihmeet

Description: Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014)

Ref Expression
Hypotheses dihmeet.b 
|- B = ( Base ` K )
dihmeet.m 
|- ./\ = ( meet ` K )
dihmeet.h 
|- H = ( LHyp ` K )
dihmeet.i 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihmeet 
|- ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )

Proof

Step Hyp Ref Expression
1 dihmeet.b 
 |-  B = ( Base ` K )
2 dihmeet.m 
 |-  ./\ = ( meet ` K )
3 dihmeet.h 
 |-  H = ( LHyp ` K )
4 dihmeet.i 
 |-  I = ( ( DIsoH ` K ) ` W )
5 eqid 
 |-  ( glb ` K ) = ( glb ` K )
6 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> K e. HL )
7 simp2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> X e. B )
8 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> Y e. B )
9 5 2 6 7 8 meetval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( ( glb ` K ) ` { X , Y } ) )
10 9 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( I ` ( X ./\ Y ) ) = ( I ` ( ( glb ` K ) ` { X , Y } ) ) )
11 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( K e. HL /\ W e. H ) )
12 prssi 
 |-  ( ( X e. B /\ Y e. B ) -> { X , Y } C_ B )
13 12 3adant1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> { X , Y } C_ B )
14 prnzg 
 |-  ( X e. B -> { X , Y } =/= (/) )
15 14 3ad2ant2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> { X , Y } =/= (/) )
16 1 5 3 4 dihglb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( { X , Y } C_ B /\ { X , Y } =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( I ` x ) )
17 11 13 15 16 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( I ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( I ` x ) )
18 fveq2 
 |-  ( x = X -> ( I ` x ) = ( I ` X ) )
19 fveq2 
 |-  ( x = Y -> ( I ` x ) = ( I ` Y ) )
20 18 19 iinxprg 
 |-  ( ( X e. B /\ Y e. B ) -> |^|_ x e. { X , Y } ( I ` x ) = ( ( I ` X ) i^i ( I ` Y ) ) )
21 20 3adant1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> |^|_ x e. { X , Y } ( I ` x ) = ( ( I ` X ) i^i ( I ` Y ) ) )
22 10 17 21 3eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )