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Metamath Proof Explorer

Theorem dihmeetlem11N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem9.b 
|- B = ( Base ` K )
dihmeetlem9.l 
|- .<_ = ( le ` K )
dihmeetlem9.h 
|- H = ( LHyp ` K )
dihmeetlem9.j 
|- .\/ = ( join ` K )
dihmeetlem9.m 
|- ./\ = ( meet ` K )
dihmeetlem9.a 
|- A = ( Atoms ` K )
dihmeetlem9.u 
|- U = ( ( DVecH ` K ) ` W )
dihmeetlem9.s 
|- .(+) = ( LSSum ` U )
dihmeetlem9.i 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihmeetlem11N 
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( ( I ` ( ( X ./\ Y ) .\/ p ) ) i^i ( I ` Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )

Proof

Step Hyp Ref Expression
1 dihmeetlem9.b 
 |-  B = ( Base ` K )
2 dihmeetlem9.l 
 |-  .<_ = ( le ` K )
3 dihmeetlem9.h 
 |-  H = ( LHyp ` K )
4 dihmeetlem9.j 
 |-  .\/ = ( join ` K )
5 dihmeetlem9.m 
 |-  ./\ = ( meet ` K )
6 dihmeetlem9.a 
 |-  A = ( Atoms ` K )
7 dihmeetlem9.u 
 |-  U = ( ( DVecH ` K ) ` W )
8 dihmeetlem9.s 
 |-  .(+) = ( LSSum ` U )
9 dihmeetlem9.i 
 |-  I = ( ( DIsoH ` K ) ` W )
10 1 2 3 4 5 6 7 8 9 dihmeetlem10N 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( I ` ( ( X ./\ Y ) .\/ p ) ) = ( ( I ` X ) i^i ( I ` ( Y .\/ p ) ) ) )
11 10 ineq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( ( I ` ( ( X ./\ Y ) .\/ p ) ) i^i ( I ` Y ) ) = ( ( ( I ` X ) i^i ( I ` ( Y .\/ p ) ) ) i^i ( I ` Y ) ) )
12 inass 
 |-  ( ( ( I ` X ) i^i ( I ` ( Y .\/ p ) ) ) i^i ( I ` Y ) ) = ( ( I ` X ) i^i ( ( I ` ( Y .\/ p ) ) i^i ( I ` Y ) ) )
13 simpl1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> K e. HL )
14 13 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> K e. Lat )
15 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> Y e. B )
16 simprll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> p e. A )
17 1 6 atbase 
 |-  ( p e. A -> p e. B )
18 16 17 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> p e. B )
19 1 2 4 latlej1 
 |-  ( ( K e. Lat /\ Y e. B /\ p e. B ) -> Y .<_ ( Y .\/ p ) )
20 14 15 18 19 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> Y .<_ ( Y .\/ p ) )
21 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( K e. HL /\ W e. H ) )
22 1 4 latjcl 
 |-  ( ( K e. Lat /\ Y e. B /\ p e. B ) -> ( Y .\/ p ) e. B )
23 14 15 18 22 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( Y .\/ p ) e. B )
24 1 2 3 9 dihord 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Y e. B /\ ( Y .\/ p ) e. B ) -> ( ( I ` Y ) C_ ( I ` ( Y .\/ p ) ) <-> Y .<_ ( Y .\/ p ) ) )
25 21 15 23 24 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( ( I ` Y ) C_ ( I ` ( Y .\/ p ) ) <-> Y .<_ ( Y .\/ p ) ) )
26 20 25 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( I ` Y ) C_ ( I ` ( Y .\/ p ) ) )
27 sseqin2 
 |-  ( ( I ` Y ) C_ ( I ` ( Y .\/ p ) ) <-> ( ( I ` ( Y .\/ p ) ) i^i ( I ` Y ) ) = ( I ` Y ) )
28 26 27 sylib 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( ( I ` ( Y .\/ p ) ) i^i ( I ` Y ) ) = ( I ` Y ) )
29 28 ineq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( ( I ` X ) i^i ( ( I ` ( Y .\/ p ) ) i^i ( I ` Y ) ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )
30 12 29 eqtrid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( ( ( I ` X ) i^i ( I ` ( Y .\/ p ) ) ) i^i ( I ` Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )
31 11 30 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X ) ) -> ( ( I ` ( ( X ./\ Y ) .\/ p ) ) i^i ( I ` Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )