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Metamath Proof Explorer

Theorem dihjatcc

Description: Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014)

Ref Expression
Hypotheses dihjatcc.l 
|- .<_ = ( le ` K )
dihjatcc.h 
|- H = ( LHyp ` K )
dihjatcc.j 
|- .\/ = ( join ` K )
dihjatcc.a 
|- A = ( Atoms ` K )
dihjatcc.u 
|- U = ( ( DVecH ` K ) ` W )
dihjatcc.s 
|- .(+) = ( LSSum ` U )
dihjatcc.i 
|- I = ( ( DIsoH ` K ) ` W )
dihjatcc.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dihjatcc.p 
|- ( ph -> ( P e. A /\ -. P .<_ W ) )
dihjatcc.q 
|- ( ph -> ( Q e. A /\ -. Q .<_ W ) )
Assertion dihjatcc 
|- ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) )

Proof

Step Hyp Ref Expression
1 dihjatcc.l 
 |-  .<_ = ( le ` K )
2 dihjatcc.h 
 |-  H = ( LHyp ` K )
3 dihjatcc.j 
 |-  .\/ = ( join ` K )
4 dihjatcc.a 
 |-  A = ( Atoms ` K )
5 dihjatcc.u 
 |-  U = ( ( DVecH ` K ) ` W )
6 dihjatcc.s 
 |-  .(+) = ( LSSum ` U )
7 dihjatcc.i 
 |-  I = ( ( DIsoH ` K ) ` W )
8 dihjatcc.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
9 dihjatcc.p 
 |-  ( ph -> ( P e. A /\ -. P .<_ W ) )
10 dihjatcc.q 
 |-  ( ph -> ( Q e. A /\ -. Q .<_ W ) )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 eqid 
 |-  ( meet ` K ) = ( meet ` K )
13 eqid 
 |-  ( ( P .\/ Q ) ( meet ` K ) W ) = ( ( P .\/ Q ) ( meet ` K ) W )
14 eqid 
 |-  ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
15 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
16 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
17 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
18 eqid 
 |-  ( iota_ d e. ( ( LTrn ` K ) ` W ) ( d ` ( ( oc ` K ) ` W ) ) = P ) = ( iota_ d e. ( ( LTrn ` K ) ` W ) ( d ` ( ( oc ` K ) ` W ) ) = P )
19 eqid 
 |-  ( iota_ d e. ( ( LTrn ` K ) ` W ) ( d ` ( ( oc ` K ) ` W ) ) = Q ) = ( iota_ d e. ( ( LTrn ` K ) ` W ) ( d ` ( ( oc ` K ) ` W ) ) = Q )
20 eqid 
 |-  ( a e. ( ( TEndo ` K ) ` W ) |-> ( d e. ( ( LTrn ` K ) ` W ) |-> `' ( a ` d ) ) ) = ( a e. ( ( TEndo ` K ) ` W ) |-> ( d e. ( ( LTrn ` K ) ` W ) |-> `' ( a ` d ) ) )
21 eqid 
 |-  ( d e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) = ( d e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) )
22 eqid 
 |-  ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( d e. ( ( LTrn ` K ) ` W ) |-> ( ( a ` d ) o. ( b ` d ) ) ) ) = ( a e. ( ( TEndo ` K ) ` W ) , b e. ( ( TEndo ` K ) ` W ) |-> ( d e. ( ( LTrn ` K ) ` W ) |-> ( ( a ` d ) o. ( b ` d ) ) ) )
23 11 1 2 3 12 4 5 6 7 13 8 9 10 14 15 16 17 18 19 20 21 22 dihjatcclem4 
 |-  ( ph -> ( I ` ( ( P .\/ Q ) ( meet ` K ) W ) ) C_ ( ( I ` P ) .(+) ( I ` Q ) ) )
24 11 1 2 3 12 4 5 6 7 13 8 9 10 23 dihjatcclem2 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) )