Metamath Proof Explorer

Theorem dihglblem5aN

Description: A conjunction property of isomorphism H. (Contributed by NM, 21-Mar-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihglblem5a.b 
|- B = ( Base ` K )
dihglblem5a.m 
|- ./\ = ( meet ` K )
dihglblem5a.h 
|- H = ( LHyp ` K )
dihglblem5a.i 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihglblem5aN 
|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( I ` ( X ./\ W ) ) = ( ( I ` X ) i^i ( I ` W ) ) )

Proof

Step Hyp Ref Expression
1 dihglblem5a.b 
 |-  B = ( Base ` K )
2 dihglblem5a.m 
 |-  ./\ = ( meet ` K )
3 dihglblem5a.h 
 |-  H = ( LHyp ` K )
4 dihglblem5a.i 
 |-  I = ( ( DIsoH ` K ) ` W )
5 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> X ( le ` K ) W )
6 hllat 
 |-  ( K e. HL -> K e. Lat )
7 6 ad3antrrr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> K e. Lat )
8 simplr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> X e. B )
9 1 3 lhpbase 
 |-  ( W e. H -> W e. B )
10 9 ad3antlr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> W e. B )
11 eqid 
 |-  ( le ` K ) = ( le ` K )
12 1 11 2 latleeqm1 
 |-  ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ( le ` K ) W <-> ( X ./\ W ) = X ) )
13 7 8 10 12 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> ( X ( le ` K ) W <-> ( X ./\ W ) = X ) )
14 5 13 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> ( X ./\ W ) = X )
15 14 fveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> ( I ` ( X ./\ W ) ) = ( I ` X ) )
16 simpll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> ( K e. HL /\ W e. H ) )
17 1 11 3 4 dihord 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ W e. B ) -> ( ( I ` X ) C_ ( I ` W ) <-> X ( le ` K ) W ) )
18 16 8 10 17 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> ( ( I ` X ) C_ ( I ` W ) <-> X ( le ` K ) W ) )
19 5 18 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> ( I ` X ) C_ ( I ` W ) )
20 df-ss 
 |-  ( ( I ` X ) C_ ( I ` W ) <-> ( ( I ` X ) i^i ( I ` W ) ) = ( I ` X ) )
21 19 20 sylib 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> ( ( I ` X ) i^i ( I ` W ) ) = ( I ` X ) )
22 15 21 eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ X ( le ` K ) W ) -> ( I ` ( X ./\ W ) ) = ( ( I ` X ) i^i ( I ` W ) ) )
23 eqid 
 |-  ( join ` K ) = ( join ` K )
24 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
25 eqid 
 |-  ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
26 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
27 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
28 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
29 eqid 
 |-  ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = q ) = ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = q )
30 eqid 
 |-  ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
31 1 2 3 4 11 23 24 25 26 27 28 29 30 dihglblem5apreN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X ( le ` K ) W ) ) -> ( I ` ( X ./\ W ) ) = ( ( I ` X ) i^i ( I ` W ) ) )
32 31 anassrs 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B ) /\ -. X ( le ` K ) W ) -> ( I ` ( X ./\ W ) ) = ( ( I ` X ) i^i ( I ` W ) ) )
33 22 32 pm2.61dan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( I ` ( X ./\ W ) ) = ( ( I ` X ) i^i ( I ` W ) ) )