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Metamath Proof Explorer

Theorem dihjatb

Description: Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014)

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

Proof

Step Hyp Ref Expression
1 dihjatb.l 
 |-  .<_ = ( le ` K )
2 dihjatb.h 
 |-  H = ( LHyp ` K )
3 dihjatb.j 
 |-  .\/ = ( join ` K )
4 dihjatb.a 
 |-  A = ( Atoms ` K )
5 dihjatb.u 
 |-  U = ( ( DVecH ` K ) ` W )
6 dihjatb.s 
 |-  .(+) = ( LSSum ` U )
7 dihjatb.i 
 |-  I = ( ( DIsoH ` K ) ` W )
8 dihjatb.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
9 dihjatb.p 
 |-  ( ph -> ( P e. A /\ P .<_ W ) )
10 dihjatb.q 
 |-  ( ph -> ( Q e. A /\ Q .<_ W ) )
11 1 3 4 2 5 6 7 8 9 10 dih2dimb 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) C_ ( ( I ` P ) .(+) ( I ` Q ) ) )
12 eqid 
 |-  ( Base ` K ) = ( Base ` K )
13 9 simpld 
 |-  ( ph -> P e. A )
14 12 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
15 13 14 syl 
 |-  ( ph -> P e. ( Base ` K ) )
16 10 simpld 
 |-  ( ph -> Q e. A )
17 12 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
18 16 17 syl 
 |-  ( ph -> Q e. ( Base ` K ) )
19 12 2 3 5 6 7 8 15 18 dihsumssj 
 |-  ( ph -> ( ( I ` P ) .(+) ( I ` Q ) ) C_ ( I ` ( P .\/ Q ) ) )
20 11 19 eqssd 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) )