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Metamath Proof Explorer

Theorem dihmeetlem15N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem14.b 
|- B = ( Base ` K )
dihmeetlem14.l 
|- .<_ = ( le ` K )
dihmeetlem14.h 
|- H = ( LHyp ` K )
dihmeetlem14.j 
|- .\/ = ( join ` K )
dihmeetlem14.m 
|- ./\ = ( meet ` K )
dihmeetlem14.a 
|- A = ( Atoms ` K )
dihmeetlem14.u 
|- U = ( ( DVecH ` K ) ` W )
dihmeetlem14.s 
|- .(+) = ( LSSum ` U )
dihmeetlem14.i 
|- I = ( ( DIsoH ` K ) ` W )
dihmeetlem15.z 
|- .0. = ( 0g ` U )
Assertion dihmeetlem15N 
|- ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( ( I ` r ) i^i ( I ` p ) ) = { .0. } )

Proof

Step Hyp Ref Expression
1 dihmeetlem14.b 
 |-  B = ( Base ` K )
2 dihmeetlem14.l 
 |-  .<_ = ( le ` K )
3 dihmeetlem14.h 
 |-  H = ( LHyp ` K )
4 dihmeetlem14.j 
 |-  .\/ = ( join ` K )
5 dihmeetlem14.m 
 |-  ./\ = ( meet ` K )
6 dihmeetlem14.a 
 |-  A = ( Atoms ` K )
7 dihmeetlem14.u 
 |-  U = ( ( DVecH ` K ) ` W )
8 dihmeetlem14.s 
 |-  .(+) = ( LSSum ` U )
9 dihmeetlem14.i 
 |-  I = ( ( DIsoH ` K ) ` W )
10 dihmeetlem15.z 
 |-  .0. = ( 0g ` U )
11 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( K e. HL /\ W e. H ) )
12 simpr1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( r e. A /\ -. r .<_ W ) )
13 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( p e. A /\ -. p .<_ W ) )
14 simpl3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> -. p .<_ W )
15 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> r = p )
16 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> r .<_ Y )
17 15 16 eqbrtrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> p .<_ Y )
18 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> K e. HL )
19 18 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> K e. Lat )
20 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> p e. A )
21 1 6 atbase 
 |-  ( p e. A -> p e. B )
22 20 21 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> p e. B )
23 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> Y e. B )
24 1 2 5 latleeqm2 
 |-  ( ( K e. Lat /\ p e. B /\ Y e. B ) -> ( p .<_ Y <-> ( Y ./\ p ) = p ) )
25 19 22 23 24 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> ( p .<_ Y <-> ( Y ./\ p ) = p ) )
26 17 25 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> ( Y ./\ p ) = p )
27 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> ( Y ./\ p ) .<_ W )
28 26 27 eqbrtrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) /\ r = p ) -> p .<_ W )
29 28 3expia 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( r = p -> p .<_ W ) )
30 29 necon3bd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( -. p .<_ W -> r =/= p ) )
31 14 30 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> r =/= p )
32 eqid 
 |-  ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
33 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
34 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
35 eqid 
 |-  ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
36 eqid 
 |-  ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = r ) = ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = r )
37 eqid 
 |-  ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = p ) = ( iota_ h e. ( ( LTrn ` K ) ` W ) ( h ` ( ( oc ` K ) ` W ) ) = p )
38 1 2 4 6 3 32 33 34 35 9 7 10 36 37 dihmeetlem13N 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( r e. A /\ -. r .<_ W ) /\ ( p e. A /\ -. p .<_ W ) ) /\ r =/= p ) -> ( ( I ` r ) i^i ( I ` p ) ) = { .0. } )
39 11 12 13 31 38 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( p e. A /\ -. p .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ r .<_ Y /\ ( Y ./\ p ) .<_ W ) ) -> ( ( I ` r ) i^i ( I ` p ) ) = { .0. } )