Metamath Proof Explorer

Theorem dihmeetlem18N

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 )
dihmeetlem18.z 
|- .0. = ( 0g ` U )
Assertion dihmeetlem18N 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( ( I ` Y ) 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 dihmeetlem18.z 
 |-  .0. = ( 0g ` U )
11 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
12 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( X e. B /\ -. X .<_ W ) )
13 simpr1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( p e. A /\ -. p .<_ W ) )
14 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> Y e. B )
15 simpr33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( X ./\ Y ) .<_ W )
16 simpr31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> p .<_ X )
17 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
18 1 2 3 4 5 6 7 8 9 17 dihmeetlem17N 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( p e. A /\ -. p .<_ W ) ) /\ ( Y e. B /\ ( X ./\ Y ) .<_ W /\ p .<_ X ) ) -> ( Y ./\ p ) = ( 0. ` K ) )
19 11 12 13 14 15 16 18 syl33anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( Y ./\ p ) = ( 0. ` K ) )
20 19 fveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( I ` ( Y ./\ p ) ) = ( I ` ( 0. ` K ) ) )
21 simpr2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( r e. A /\ -. r .<_ W ) )
22 simpr32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> r .<_ Y )
23 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> K e. HL )
24 hlop 
 |-  ( K e. HL -> K e. OP )
25 23 24 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> K e. OP )
26 simpl1r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> W e. H )
27 1 3 lhpbase 
 |-  ( W e. H -> W e. B )
28 26 27 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> W e. B )
29 1 2 17 op0le 
 |-  ( ( K e. OP /\ W e. B ) -> ( 0. ` K ) .<_ W )
30 25 28 29 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( 0. ` K ) .<_ W )
31 19 30 eqbrtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( Y ./\ p ) .<_ W )
32 1 2 3 4 5 6 7 8 9 dihmeetlem16N 
 |-  ( ( ( ( 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 ` ( Y ./\ p ) ) = ( ( I ` Y ) i^i ( I ` p ) ) )
33 11 14 13 21 22 31 32 syl33anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( I ` ( Y ./\ p ) ) = ( ( I ` Y ) i^i ( I ` p ) ) )
34 17 3 9 7 10 dih0 
 |-  ( ( K e. HL /\ W e. H ) -> ( I ` ( 0. ` K ) ) = { .0. } )
35 11 34 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( I ` ( 0. ` K ) ) = { .0. } )
36 20 33 35 3eqtr3d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ ( r e. A /\ -. r .<_ W ) /\ ( p .<_ X /\ r .<_ Y /\ ( X ./\ Y ) .<_ W ) ) ) -> ( ( I ` Y ) i^i ( I ` p ) ) = { .0. } )