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 ) )`
` |-  ( ( X e. B /\ Y e. B ) -> { X , Y } C_ B )`
` |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> { X , Y } C_ B )`
14 prnzg
` |-  ( X e. B -> { X , Y } =/= (/) )`
` |-  ( ( ( 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 ) ) )`
` |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> |^|_ x e. { X , Y } ( I ` x ) = ( ( I ` X ) i^i ( I ` Y ) ) )`
` |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )`