Metamath Proof Explorer

Theorem dihjatcclem2

Description: Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014)

Ref Expression
Hypotheses dihjatcclem.b 
|- B = ( Base ` K )
dihjatcclem.l 
|- .<_ = ( le ` K )
dihjatcclem.h 
|- H = ( LHyp ` K )
dihjatcclem.j 
|- .\/ = ( join ` K )
dihjatcclem.m 
|- ./\ = ( meet ` K )
dihjatcclem.a 
|- A = ( Atoms ` K )
dihjatcclem.u 
|- U = ( ( DVecH ` K ) ` W )
dihjatcclem.s 
|- .(+) = ( LSSum ` U )
dihjatcclem.i 
|- I = ( ( DIsoH ` K ) ` W )
dihjatcclem.v 
|- V = ( ( P .\/ Q ) ./\ W )
dihjatcclem.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dihjatcclem.p 
|- ( ph -> ( P e. A /\ -. P .<_ W ) )
dihjatcclem.q 
|- ( ph -> ( Q e. A /\ -. Q .<_ W ) )
dihjatcclem2.c 
|- ( ph -> ( I ` V ) C_ ( ( I ` P ) .(+) ( I ` Q ) ) )
Assertion dihjatcclem2 
|- ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) )

Proof

Step Hyp Ref Expression
1 dihjatcclem.b 
 |-  B = ( Base ` K )
2 dihjatcclem.l 
 |-  .<_ = ( le ` K )
3 dihjatcclem.h 
 |-  H = ( LHyp ` K )
4 dihjatcclem.j 
 |-  .\/ = ( join ` K )
5 dihjatcclem.m 
 |-  ./\ = ( meet ` K )
6 dihjatcclem.a 
 |-  A = ( Atoms ` K )
7 dihjatcclem.u 
 |-  U = ( ( DVecH ` K ) ` W )
8 dihjatcclem.s 
 |-  .(+) = ( LSSum ` U )
9 dihjatcclem.i 
 |-  I = ( ( DIsoH ` K ) ` W )
10 dihjatcclem.v 
 |-  V = ( ( P .\/ Q ) ./\ W )
11 dihjatcclem.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
12 dihjatcclem.p 
 |-  ( ph -> ( P e. A /\ -. P .<_ W ) )
13 dihjatcclem.q 
 |-  ( ph -> ( Q e. A /\ -. Q .<_ W ) )
14 dihjatcclem2.c 
 |-  ( ph -> ( I ` V ) C_ ( ( I ` P ) .(+) ( I ` Q ) ) )
15 1 2 3 4 5 6 7 8 9 10 11 12 13 dihjatcclem1 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) = ( ( ( I ` P ) .(+) ( I ` Q ) ) .(+) ( I ` V ) ) )
16 3 7 11 dvhlmod 
 |-  ( ph -> U e. LMod )
17 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
18 17 lsssssubg 
 |-  ( U e. LMod -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
19 16 18 syl 
 |-  ( ph -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
20 12 simpld 
 |-  ( ph -> P e. A )
21 1 6 atbase 
 |-  ( P e. A -> P e. B )
22 20 21 syl 
 |-  ( ph -> P e. B )
23 1 3 9 7 17 dihlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. B ) -> ( I ` P ) e. ( LSubSp ` U ) )
24 11 22 23 syl2anc 
 |-  ( ph -> ( I ` P ) e. ( LSubSp ` U ) )
25 13 simpld 
 |-  ( ph -> Q e. A )
26 1 6 atbase 
 |-  ( Q e. A -> Q e. B )
27 25 26 syl 
 |-  ( ph -> Q e. B )
28 1 3 9 7 17 dihlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Q e. B ) -> ( I ` Q ) e. ( LSubSp ` U ) )
29 11 27 28 syl2anc 
 |-  ( ph -> ( I ` Q ) e. ( LSubSp ` U ) )
30 17 8 lsmcl 
 |-  ( ( U e. LMod /\ ( I ` P ) e. ( LSubSp ` U ) /\ ( I ` Q ) e. ( LSubSp ` U ) ) -> ( ( I ` P ) .(+) ( I ` Q ) ) e. ( LSubSp ` U ) )
31 16 24 29 30 syl3anc 
 |-  ( ph -> ( ( I ` P ) .(+) ( I ` Q ) ) e. ( LSubSp ` U ) )
32 19 31 sseldd 
 |-  ( ph -> ( ( I ` P ) .(+) ( I ` Q ) ) e. ( SubGrp ` U ) )
33 10 fveq2i 
 |-  ( I ` V ) = ( I ` ( ( P .\/ Q ) ./\ W ) )
34 11 simpld 
 |-  ( ph -> K e. HL )
35 34 hllatd 
 |-  ( ph -> K e. Lat )
36 1 4 6 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. B )
37 34 20 25 36 syl3anc 
 |-  ( ph -> ( P .\/ Q ) e. B )
38 11 simprd 
 |-  ( ph -> W e. H )
39 1 3 lhpbase 
 |-  ( W e. H -> W e. B )
40 38 39 syl 
 |-  ( ph -> W e. B )
41 1 5 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. B /\ W e. B ) -> ( ( P .\/ Q ) ./\ W ) e. B )
42 35 37 40 41 syl3anc 
 |-  ( ph -> ( ( P .\/ Q ) ./\ W ) e. B )
43 1 3 9 7 17 dihlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P .\/ Q ) ./\ W ) e. B ) -> ( I ` ( ( P .\/ Q ) ./\ W ) ) e. ( LSubSp ` U ) )
44 11 42 43 syl2anc 
 |-  ( ph -> ( I ` ( ( P .\/ Q ) ./\ W ) ) e. ( LSubSp ` U ) )
45 33 44 eqeltrid 
 |-  ( ph -> ( I ` V ) e. ( LSubSp ` U ) )
46 19 45 sseldd 
 |-  ( ph -> ( I ` V ) e. ( SubGrp ` U ) )
47 8 lsmss2 
 |-  ( ( ( ( I ` P ) .(+) ( I ` Q ) ) e. ( SubGrp ` U ) /\ ( I ` V ) e. ( SubGrp ` U ) /\ ( I ` V ) C_ ( ( I ` P ) .(+) ( I ` Q ) ) ) -> ( ( ( I ` P ) .(+) ( I ` Q ) ) .(+) ( I ` V ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) )
48 32 46 14 47 syl3anc 
 |-  ( ph -> ( ( ( I ` P ) .(+) ( I ` Q ) ) .(+) ( I ` V ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) )
49 15 48 eqtrd 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) )