Metamath Proof Explorer

Theorem dihmeetlem12N

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 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 ) ) )

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 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( K e. HL /\ W e. H ) )
11 simpl2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> X e. B )
12 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> Y e. B )
13 simpr1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( p e. A /\ -. p .<_ W ) )
14 simpr2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> p .<_ X )
15 simpr3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( X ./\ Y ) .<_ W )
16 1 2 3 4 5 6 7 8 9 dihmeetlem8N 
 |-  ( ( ( ( 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 ) .\/ p ) ) = ( ( I ` p ) .(+) ( I ` ( X ./\ Y ) ) ) )
17 10 11 12 13 14 15 16 syl312anc 
 |-  ( ( ( ( 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 ) .\/ p ) ) = ( ( I ` p ) .(+) ( I ` ( X ./\ Y ) ) ) )
18 17 ineq1d 
 |-  ( ( ( ( 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 ) .\/ p ) ) i^i ( I ` Y ) ) = ( ( ( I ` p ) .(+) ( I ` ( X ./\ Y ) ) ) i^i ( I ` Y ) ) )
19 1 2 3 4 5 6 7 8 9 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 ) ) )
20 19 3adantr3 
 |-  ( ( ( ( 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 ) .\/ p ) ) i^i ( I ` Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )
21 simpr1l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> p e. A )
22 1 2 3 4 5 6 7 8 9 dihmeetlem9N 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ Y e. B ) /\ p e. A ) -> ( ( ( I ` p ) .(+) ( I ` ( X ./\ Y ) ) ) i^i ( I ` Y ) ) = ( ( I ` ( X ./\ Y ) ) .(+) ( ( I ` p ) i^i ( I ` Y ) ) ) )
23 10 11 12 21 22 syl121anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( ( p e. A /\ -. p .<_ W ) /\ p .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( ( ( I ` p ) .(+) ( I ` ( X ./\ Y ) ) ) i^i ( I ` Y ) ) = ( ( I ` ( X ./\ Y ) ) .(+) ( ( I ` p ) i^i ( I ` Y ) ) ) )
24 18 20 23 3eqtr3rd 
 |-  ( ( ( ( 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 ) ) )