Metamath Proof Explorer

Theorem pmapmeet

Description: The projective map of a meet. (Contributed by NM, 25-Jan-2012)

Ref Expression
Hypotheses pmapmeet.b 
|- B = ( Base ` K )
pmapmeet.m 
|- ./\ = ( meet ` K )
pmapmeet.a 
|- A = ( Atoms ` K )
pmapmeet.p 
|- P = ( pmap ` K )
Assertion pmapmeet 
|- ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( X ./\ Y ) ) = ( ( P ` X ) i^i ( P ` Y ) ) )

Proof

Step Hyp Ref Expression
1 pmapmeet.b 
 |-  B = ( Base ` K )
2 pmapmeet.m 
 |-  ./\ = ( meet ` K )
3 pmapmeet.a 
 |-  A = ( Atoms ` K )
4 pmapmeet.p 
 |-  P = ( pmap ` K )
5 eqid 
 |-  ( glb ` K ) = ( glb ` K )
6 simp1 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> K e. HL )
7 simp2 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> X e. B )
8 simp3 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> Y e. B )
9 5 2 6 7 8 meetval 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) = ( ( glb ` K ) ` { X , Y } ) )
10 9 fveq2d 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( X ./\ Y ) ) = ( P ` ( ( glb ` K ) ` { X , Y } ) ) )
11 prssi 
 |-  ( ( X e. B /\ Y e. B ) -> { X , Y } C_ B )
12 11 3adant1 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> { X , Y } C_ B )
13 prnzg 
 |-  ( X e. B -> { X , Y } =/= (/) )
14 13 3ad2ant2 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> { X , Y } =/= (/) )
15 1 5 4 pmapglb 
 |-  ( ( K e. HL /\ { X , Y } C_ B /\ { X , Y } =/= (/) ) -> ( P ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( P ` x ) )
16 6 12 14 15 syl3anc 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( P ` x ) )
17 fveq2 
 |-  ( x = X -> ( P ` x ) = ( P ` X ) )
18 fveq2 
 |-  ( x = Y -> ( P ` x ) = ( P ` Y ) )
19 17 18 iinxprg 
 |-  ( ( X e. B /\ Y e. B ) -> |^|_ x e. { X , Y } ( P ` x ) = ( ( P ` X ) i^i ( P ` Y ) ) )
20 19 3adant1 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> |^|_ x e. { X , Y } ( P ` x ) = ( ( P ` X ) i^i ( P ` Y ) ) )
21 10 16 20 3eqtrd 
 |-  ( ( K e. HL /\ X e. B /\ Y e. B ) -> ( P ` ( X ./\ Y ) ) = ( ( P ` X ) i^i ( P ` Y ) ) )