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Metamath Proof Explorer

Theorem dihmeetlem19N

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 )
Assertion dihmeetlem19N 
|- ( ( ( ( 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 ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )

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 incom 
 |-  ( ( I ` p ) i^i ( I ` Y ) ) = ( ( I ` Y ) i^i ( I ` p ) )
11 eqid 
 |-  ( 0g ` U ) = ( 0g ` U )
12 1 2 3 4 5 6 7 8 9 11 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 ) ) = { ( 0g ` U ) } )
13 10 12 eqtrid 
 |-  ( ( ( ( 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 ` p ) i^i ( I ` Y ) ) = { ( 0g ` U ) } )
14 13 oveq2d 
 |-  ( ( ( ( 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 ` ( X ./\ Y ) ) .(+) ( ( I ` p ) i^i ( I ` Y ) ) ) = ( ( I ` ( X ./\ Y ) ) .(+) { ( 0g ` U ) } ) )
15 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 ) )
16 simpl2l 
 |-  ( ( ( ( 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 )
17 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 )
18 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 ) )
19 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 )
20 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 )
21 1 2 3 4 5 6 7 8 9 dihmeetlem12N 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( ( I ` ( X ./\ Y ) ) .(+) ( ( I ` p ) i^i ( I ` Y ) ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )
22 15 16 17 18 19 20 21 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 ` ( X ./\ Y ) ) .(+) ( ( I ` p ) i^i ( I ` Y ) ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )
23 3 7 15 dvhlmod 
 |-  ( ( ( ( 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 ) ) ) -> U e. LMod )
24 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 )
25 24 hllatd 
 |-  ( ( ( ( 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. Lat )
26 1 5 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )
27 25 16 17 26 syl3anc 
 |-  ( ( ( ( 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 ) e. B )
28 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
29 1 3 9 7 28 dihlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X ./\ Y ) e. B ) -> ( I ` ( X ./\ Y ) ) e. ( LSubSp ` U ) )
30 15 27 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 ) ) ) -> ( I ` ( X ./\ Y ) ) e. ( LSubSp ` U ) )
31 28 lsssubg 
 |-  ( ( U e. LMod /\ ( I ` ( X ./\ Y ) ) e. ( LSubSp ` U ) ) -> ( I ` ( X ./\ Y ) ) e. ( SubGrp ` U ) )
32 23 30 31 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 ) ) ) -> ( I ` ( X ./\ Y ) ) e. ( SubGrp ` U ) )
33 11 8 lsm01 
 |-  ( ( I ` ( X ./\ Y ) ) e. ( SubGrp ` U ) -> ( ( I ` ( X ./\ Y ) ) .(+) { ( 0g ` U ) } ) = ( I ` ( X ./\ Y ) ) )
34 32 33 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 ` ( X ./\ Y ) ) .(+) { ( 0g ` U ) } ) = ( I ` ( X ./\ Y ) ) )
35 14 22 34 3eqtr3rd 
 |-  ( ( ( ( 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 ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )