Metamath Proof Explorer

Theorem dih2dimb

Description: Extend dib2dim to isomorphism H. (Contributed by NM, 22-Sep-2014)

Ref Expression
Hypotheses dih2dimb.l 
|- .<_ = ( le ` K )
dih2dimb.j 
|- .\/ = ( join ` K )
dih2dimb.a 
|- A = ( Atoms ` K )
dih2dimb.h 
|- H = ( LHyp ` K )
dih2dimb.u 
|- U = ( ( DVecH ` K ) ` W )
dih2dimb.s 
|- .(+) = ( LSSum ` U )
dih2dimb.i 
|- I = ( ( DIsoH ` K ) ` W )
dih2dimb.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dih2dimb.p 
|- ( ph -> ( P e. A /\ P .<_ W ) )
dih2dimb.q 
|- ( ph -> ( Q e. A /\ Q .<_ W ) )
Assertion dih2dimb 
|- ( ph -> ( I ` ( P .\/ Q ) ) C_ ( ( I ` P ) .(+) ( I ` Q ) ) )

Proof

Step Hyp Ref Expression
1 dih2dimb.l 
 |-  .<_ = ( le ` K )
2 dih2dimb.j 
 |-  .\/ = ( join ` K )
3 dih2dimb.a 
 |-  A = ( Atoms ` K )
4 dih2dimb.h 
 |-  H = ( LHyp ` K )
5 dih2dimb.u 
 |-  U = ( ( DVecH ` K ) ` W )
6 dih2dimb.s 
 |-  .(+) = ( LSSum ` U )
7 dih2dimb.i 
 |-  I = ( ( DIsoH ` K ) ` W )
8 dih2dimb.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
9 dih2dimb.p 
 |-  ( ph -> ( P e. A /\ P .<_ W ) )
10 dih2dimb.q 
 |-  ( ph -> ( Q e. A /\ Q .<_ W ) )
11 eqid 
 |-  ( ( DIsoB ` K ) ` W ) = ( ( DIsoB ` K ) ` W )
12 1 2 3 4 5 6 11 8 9 10 dib2dim 
 |-  ( ph -> ( ( ( DIsoB ` K ) ` W ) ` ( P .\/ Q ) ) C_ ( ( ( ( DIsoB ` K ) ` W ) ` P ) .(+) ( ( ( DIsoB ` K ) ` W ) ` Q ) ) )
13 8 simpld 
 |-  ( ph -> K e. HL )
14 9 simpld 
 |-  ( ph -> P e. A )
15 10 simpld 
 |-  ( ph -> Q e. A )
16 eqid 
 |-  ( Base ` K ) = ( Base ` K )
17 16 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
18 13 14 15 17 syl3anc 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
19 9 simprd 
 |-  ( ph -> P .<_ W )
20 10 simprd 
 |-  ( ph -> Q .<_ W )
21 13 hllatd 
 |-  ( ph -> K e. Lat )
22 16 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
23 14 22 syl 
 |-  ( ph -> P e. ( Base ` K ) )
24 16 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
25 15 24 syl 
 |-  ( ph -> Q e. ( Base ` K ) )
26 8 simprd 
 |-  ( ph -> W e. H )
27 16 4 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
28 26 27 syl 
 |-  ( ph -> W e. ( Base ` K ) )
29 16 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( P .<_ W /\ Q .<_ W ) <-> ( P .\/ Q ) .<_ W ) )
30 21 23 25 28 29 syl13anc 
 |-  ( ph -> ( ( P .<_ W /\ Q .<_ W ) <-> ( P .\/ Q ) .<_ W ) )
31 19 20 30 mpbi2and 
 |-  ( ph -> ( P .\/ Q ) .<_ W )
32 16 1 4 7 11 dihvalb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( P .\/ Q ) .<_ W ) ) -> ( I ` ( P .\/ Q ) ) = ( ( ( DIsoB ` K ) ` W ) ` ( P .\/ Q ) ) )
33 8 18 31 32 syl12anc 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) = ( ( ( DIsoB ` K ) ` W ) ` ( P .\/ Q ) ) )
34 16 1 4 7 11 dihvalb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. ( Base ` K ) /\ P .<_ W ) ) -> ( I ` P ) = ( ( ( DIsoB ` K ) ` W ) ` P ) )
35 8 23 19 34 syl12anc 
 |-  ( ph -> ( I ` P ) = ( ( ( DIsoB ` K ) ` W ) ` P ) )
36 16 1 4 7 11 dihvalb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. ( Base ` K ) /\ Q .<_ W ) ) -> ( I ` Q ) = ( ( ( DIsoB ` K ) ` W ) ` Q ) )
37 8 25 20 36 syl12anc 
 |-  ( ph -> ( I ` Q ) = ( ( ( DIsoB ` K ) ` W ) ` Q ) )
38 35 37 oveq12d 
 |-  ( ph -> ( ( I ` P ) .(+) ( I ` Q ) ) = ( ( ( ( DIsoB ` K ) ` W ) ` P ) .(+) ( ( ( DIsoB ` K ) ` W ) ` Q ) ) )
39 12 33 38 3sstr4d 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) C_ ( ( I ` P ) .(+) ( I ` Q ) ) )