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Metamath Proof Explorer

Theorem dihjatcclem1

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 ) )
Assertion dihjatcclem1 
|- ( ph -> ( I ` ( P .\/ Q ) ) = ( ( ( I ` P ) .(+) ( I ` Q ) ) .(+) ( I ` V ) ) )

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 3 7 11 dvhlmod 
 |-  ( ph -> U e. LMod )
15 lmodabl 
 |-  ( U e. LMod -> U e. Abel )
16 14 15 syl 
 |-  ( ph -> U e. Abel )
17 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
18 17 lsssssubg 
 |-  ( U e. LMod -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
19 14 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 19 24 sseldd 
 |-  ( ph -> ( I ` P ) e. ( SubGrp ` U ) )
26 11 simpld 
 |-  ( ph -> K e. HL )
27 26 hllatd 
 |-  ( ph -> K e. Lat )
28 13 simpld 
 |-  ( ph -> Q e. A )
29 1 4 6 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. B )
30 26 20 28 29 syl3anc 
 |-  ( ph -> ( P .\/ Q ) e. B )
31 11 simprd 
 |-  ( ph -> W e. H )
32 1 3 lhpbase 
 |-  ( W e. H -> W e. B )
33 31 32 syl 
 |-  ( ph -> W e. B )
34 1 5 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. B /\ W e. B ) -> ( ( P .\/ Q ) ./\ W ) e. B )
35 27 30 33 34 syl3anc 
 |-  ( ph -> ( ( P .\/ Q ) ./\ W ) e. B )
36 10 35 eqeltrid 
 |-  ( ph -> V e. B )
37 1 3 9 7 17 dihlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ V e. B ) -> ( I ` V ) e. ( LSubSp ` U ) )
38 11 36 37 syl2anc 
 |-  ( ph -> ( I ` V ) e. ( LSubSp ` U ) )
39 19 38 sseldd 
 |-  ( ph -> ( I ` V ) e. ( SubGrp ` U ) )
40 1 6 atbase 
 |-  ( Q e. A -> Q e. B )
41 28 40 syl 
 |-  ( ph -> Q e. B )
42 1 3 9 7 17 dihlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Q e. B ) -> ( I ` Q ) e. ( LSubSp ` U ) )
43 11 41 42 syl2anc 
 |-  ( ph -> ( I ` Q ) e. ( LSubSp ` U ) )
44 19 43 sseldd 
 |-  ( ph -> ( I ` Q ) e. ( SubGrp ` U ) )
45 8 lsm4 
 |-  ( ( U e. Abel /\ ( ( I ` P ) e. ( SubGrp ` U ) /\ ( I ` V ) e. ( SubGrp ` U ) ) /\ ( ( I ` Q ) e. ( SubGrp ` U ) /\ ( I ` V ) e. ( SubGrp ` U ) ) ) -> ( ( ( I ` P ) .(+) ( I ` V ) ) .(+) ( ( I ` Q ) .(+) ( I ` V ) ) ) = ( ( ( I ` P ) .(+) ( I ` Q ) ) .(+) ( ( I ` V ) .(+) ( I ` V ) ) ) )
46 16 25 39 44 39 45 syl122anc 
 |-  ( ph -> ( ( ( I ` P ) .(+) ( I ` V ) ) .(+) ( ( I ` Q ) .(+) ( I ` V ) ) ) = ( ( ( I ` P ) .(+) ( I ` Q ) ) .(+) ( ( I ` V ) .(+) ( I ` V ) ) ) )
47 13 simprd 
 |-  ( ph -> -. Q .<_ W )
48 47 intnand 
 |-  ( ph -> -. ( P .<_ W /\ Q .<_ W ) )
49 1 2 4 latjle12 
 |-  ( ( K e. Lat /\ ( P e. B /\ Q e. B /\ W e. B ) ) -> ( ( P .<_ W /\ Q .<_ W ) <-> ( P .\/ Q ) .<_ W ) )
50 27 22 41 33 49 syl13anc 
 |-  ( ph -> ( ( P .<_ W /\ Q .<_ W ) <-> ( P .\/ Q ) .<_ W ) )
51 48 50 mtbid 
 |-  ( ph -> -. ( P .\/ Q ) .<_ W )
52 2 4 6 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> P .<_ ( P .\/ Q ) )
53 26 20 28 52 syl3anc 
 |-  ( ph -> P .<_ ( P .\/ Q ) )
54 1 2 4 5 6 3 9 7 8 dihvalcq2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P .\/ Q ) e. B /\ -. ( P .\/ Q ) .<_ W ) /\ ( ( P e. A /\ -. P .<_ W ) /\ P .<_ ( P .\/ Q ) ) ) -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` ( ( P .\/ Q ) ./\ W ) ) ) )
55 11 30 51 12 53 54 syl122anc 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` ( ( P .\/ Q ) ./\ W ) ) ) )
56 10 fveq2i 
 |-  ( I ` V ) = ( I ` ( ( P .\/ Q ) ./\ W ) )
57 56 oveq2i 
 |-  ( ( I ` P ) .(+) ( I ` V ) ) = ( ( I ` P ) .(+) ( I ` ( ( P .\/ Q ) ./\ W ) ) )
58 55 57 eqtr4di 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` V ) ) )
59 2 4 6 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> Q .<_ ( P .\/ Q ) )
60 26 20 28 59 syl3anc 
 |-  ( ph -> Q .<_ ( P .\/ Q ) )
61 1 2 4 5 6 3 9 7 8 dihvalcq2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P .\/ Q ) e. B /\ -. ( P .\/ Q ) .<_ W ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ Q ) ) ) -> ( I ` ( P .\/ Q ) ) = ( ( I ` Q ) .(+) ( I ` ( ( P .\/ Q ) ./\ W ) ) ) )
62 11 30 51 13 60 61 syl122anc 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` Q ) .(+) ( I ` ( ( P .\/ Q ) ./\ W ) ) ) )
63 56 oveq2i 
 |-  ( ( I ` Q ) .(+) ( I ` V ) ) = ( ( I ` Q ) .(+) ( I ` ( ( P .\/ Q ) ./\ W ) ) )
64 62 63 eqtr4di 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` Q ) .(+) ( I ` V ) ) )
65 58 64 oveq12d 
 |-  ( ph -> ( ( I ` ( P .\/ Q ) ) .(+) ( I ` ( P .\/ Q ) ) ) = ( ( ( I ` P ) .(+) ( I ` V ) ) .(+) ( ( I ` Q ) .(+) ( I ` V ) ) ) )
66 1 3 9 7 17 dihlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P .\/ Q ) e. B ) -> ( I ` ( P .\/ Q ) ) e. ( LSubSp ` U ) )
67 11 30 66 syl2anc 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) e. ( LSubSp ` U ) )
68 19 67 sseldd 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) e. ( SubGrp ` U ) )
69 8 lsmidm 
 |-  ( ( I ` ( P .\/ Q ) ) e. ( SubGrp ` U ) -> ( ( I ` ( P .\/ Q ) ) .(+) ( I ` ( P .\/ Q ) ) ) = ( I ` ( P .\/ Q ) ) )
70 68 69 syl 
 |-  ( ph -> ( ( I ` ( P .\/ Q ) ) .(+) ( I ` ( P .\/ Q ) ) ) = ( I ` ( P .\/ Q ) ) )
71 65 70 eqtr3d 
 |-  ( ph -> ( ( ( I ` P ) .(+) ( I ` V ) ) .(+) ( ( I ` Q ) .(+) ( I ` V ) ) ) = ( I ` ( P .\/ Q ) ) )
72 8 lsmidm 
 |-  ( ( I ` V ) e. ( SubGrp ` U ) -> ( ( I ` V ) .(+) ( I ` V ) ) = ( I ` V ) )
73 39 72 syl 
 |-  ( ph -> ( ( I ` V ) .(+) ( I ` V ) ) = ( I ` V ) )
74 73 oveq2d 
 |-  ( ph -> ( ( ( I ` P ) .(+) ( I ` Q ) ) .(+) ( ( I ` V ) .(+) ( I ` V ) ) ) = ( ( ( I ` P ) .(+) ( I ` Q ) ) .(+) ( I ` V ) ) )
75 46 71 74 3eqtr3d 
 |-  ( ph -> ( I ` ( P .\/ Q ) ) = ( ( ( I ` P ) .(+) ( I ` Q ) ) .(+) ( I ` V ) ) )