Metamath Proof Explorer

Theorem dihmeetbN

Description: Isomorphism H of a lattice meet when one element is under the fiducial hyperplane W . (Contributed by NM, 26-Mar-2014) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 dihmeetc.b 
 |-  B = ( Base ` K )
2 dihmeetc.l 
 |-  .<_ = ( le ` K )
3 dihmeetc.m 
 |-  ./\ = ( meet ` K )
4 dihmeetc.h 
 |-  H = ( LHyp ` K )
5 dihmeetc.i 
 |-  I = ( ( DIsoH ` K ) ` W )
6 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) /\ X .<_ W ) -> ( K e. HL /\ W e. H ) )
7 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) /\ X .<_ W ) -> X e. B )
8 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) /\ X .<_ W ) -> X .<_ W )
9 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) /\ X .<_ W ) -> ( Y e. B /\ Y .<_ W ) )
10 eqid 
 |-  ( join ` K ) = ( join ` K )
11 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
12 eqid 
 |-  ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
13 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
14 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
15 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
16 eqid 
 |-  ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = q ) = ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = q )
17 eqid 
 |-  ( g e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( g e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
18 1 3 4 5 2 10 11 12 13 14 15 16 17 dihmeetlem2N 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )
19 6 7 8 9 18 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) /\ X .<_ W ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )
20 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) /\ -. X .<_ W ) -> ( K e. HL /\ W e. H ) )
21 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) /\ -. X .<_ W ) -> X e. B )
22 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) /\ -. X .<_ W ) -> -. X .<_ W )
23 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) /\ -. X .<_ W ) -> ( Y e. B /\ Y .<_ W ) )
24 1 3 4 5 2 10 11 12 13 14 15 16 17 dihmeetlem1N 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )
25 20 21 22 23 24 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) /\ -. X .<_ W ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )
26 19 25 pm2.61dan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ ( Y e. B /\ Y .<_ W ) ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )