Step |
Hyp |
Ref |
Expression |
1 |
|
dihjat.h |
|- H = ( LHyp ` K ) |
2 |
|
dihjat.j |
|- .\/ = ( join ` K ) |
3 |
|
dihjat.a |
|- A = ( Atoms ` K ) |
4 |
|
dihjat.u |
|- U = ( ( DVecH ` K ) ` W ) |
5 |
|
dihjat.s |
|- .(+) = ( LSSum ` U ) |
6 |
|
dihjat.i |
|- I = ( ( DIsoH ` K ) ` W ) |
7 |
|
dihjat.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
8 |
|
dihjat.p |
|- ( ph -> P e. A ) |
9 |
|
dihjat.q |
|- ( ph -> Q e. A ) |
10 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
11 |
7
|
adantr |
|- ( ( ph /\ ( P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( K e. HL /\ W e. H ) ) |
12 |
8
|
adantr |
|- ( ( ph /\ ( P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> P e. A ) |
13 |
|
simprl |
|- ( ( ph /\ ( P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> P ( le ` K ) W ) |
14 |
12 13
|
jca |
|- ( ( ph /\ ( P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( P e. A /\ P ( le ` K ) W ) ) |
15 |
9
|
adantr |
|- ( ( ph /\ ( P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> Q e. A ) |
16 |
|
simprr |
|- ( ( ph /\ ( P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> Q ( le ` K ) W ) |
17 |
15 16
|
jca |
|- ( ( ph /\ ( P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( Q e. A /\ Q ( le ` K ) W ) ) |
18 |
10 1 2 3 4 5 6 11 14 17
|
dihjatb |
|- ( ( ph /\ ( P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) ) |
19 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
20 |
7
|
adantr |
|- ( ( ph /\ ( P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> ( K e. HL /\ W e. H ) ) |
21 |
19 3
|
atbase |
|- ( P e. A -> P e. ( Base ` K ) ) |
22 |
8 21
|
syl |
|- ( ph -> P e. ( Base ` K ) ) |
23 |
22
|
adantr |
|- ( ( ph /\ ( P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> P e. ( Base ` K ) ) |
24 |
|
simprl |
|- ( ( ph /\ ( P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> P ( le ` K ) W ) |
25 |
23 24
|
jca |
|- ( ( ph /\ ( P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> ( P e. ( Base ` K ) /\ P ( le ` K ) W ) ) |
26 |
9
|
adantr |
|- ( ( ph /\ ( P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> Q e. A ) |
27 |
|
simprr |
|- ( ( ph /\ ( P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> -. Q ( le ` K ) W ) |
28 |
26 27
|
jca |
|- ( ( ph /\ ( P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> ( Q e. A /\ -. Q ( le ` K ) W ) ) |
29 |
19 10 1 2 3 4 5 6 20 25 28
|
dihjatc |
|- ( ( ph /\ ( P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) ) |
30 |
7
|
adantr |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( K e. HL /\ W e. H ) ) |
31 |
19 3
|
atbase |
|- ( Q e. A -> Q e. ( Base ` K ) ) |
32 |
9 31
|
syl |
|- ( ph -> Q e. ( Base ` K ) ) |
33 |
32
|
adantr |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> Q e. ( Base ` K ) ) |
34 |
|
simprr |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> Q ( le ` K ) W ) |
35 |
33 34
|
jca |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( Q e. ( Base ` K ) /\ Q ( le ` K ) W ) ) |
36 |
8
|
adantr |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> P e. A ) |
37 |
|
simprl |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> -. P ( le ` K ) W ) |
38 |
36 37
|
jca |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( P e. A /\ -. P ( le ` K ) W ) ) |
39 |
19 10 1 2 3 4 5 6 30 35 38
|
dihjatc |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( I ` ( Q .\/ P ) ) = ( ( I ` Q ) .(+) ( I ` P ) ) ) |
40 |
7
|
simpld |
|- ( ph -> K e. HL ) |
41 |
2 3
|
hlatjcom |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) ) |
42 |
40 8 9 41
|
syl3anc |
|- ( ph -> ( P .\/ Q ) = ( Q .\/ P ) ) |
43 |
42
|
fveq2d |
|- ( ph -> ( I ` ( P .\/ Q ) ) = ( I ` ( Q .\/ P ) ) ) |
44 |
43
|
adantr |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( I ` ( P .\/ Q ) ) = ( I ` ( Q .\/ P ) ) ) |
45 |
1 4 7
|
dvhlmod |
|- ( ph -> U e. LMod ) |
46 |
|
lmodabl |
|- ( U e. LMod -> U e. Abel ) |
47 |
45 46
|
syl |
|- ( ph -> U e. Abel ) |
48 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
49 |
48
|
lsssssubg |
|- ( U e. LMod -> ( LSubSp ` U ) C_ ( SubGrp ` U ) ) |
50 |
45 49
|
syl |
|- ( ph -> ( LSubSp ` U ) C_ ( SubGrp ` U ) ) |
51 |
19 1 6 4 48
|
dihlss |
|- ( ( ( K e. HL /\ W e. H ) /\ P e. ( Base ` K ) ) -> ( I ` P ) e. ( LSubSp ` U ) ) |
52 |
7 22 51
|
syl2anc |
|- ( ph -> ( I ` P ) e. ( LSubSp ` U ) ) |
53 |
50 52
|
sseldd |
|- ( ph -> ( I ` P ) e. ( SubGrp ` U ) ) |
54 |
19 1 6 4 48
|
dihlss |
|- ( ( ( K e. HL /\ W e. H ) /\ Q e. ( Base ` K ) ) -> ( I ` Q ) e. ( LSubSp ` U ) ) |
55 |
7 32 54
|
syl2anc |
|- ( ph -> ( I ` Q ) e. ( LSubSp ` U ) ) |
56 |
50 55
|
sseldd |
|- ( ph -> ( I ` Q ) e. ( SubGrp ` U ) ) |
57 |
5
|
lsmcom |
|- ( ( U e. Abel /\ ( I ` P ) e. ( SubGrp ` U ) /\ ( I ` Q ) e. ( SubGrp ` U ) ) -> ( ( I ` P ) .(+) ( I ` Q ) ) = ( ( I ` Q ) .(+) ( I ` P ) ) ) |
58 |
47 53 56 57
|
syl3anc |
|- ( ph -> ( ( I ` P ) .(+) ( I ` Q ) ) = ( ( I ` Q ) .(+) ( I ` P ) ) ) |
59 |
58
|
adantr |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( ( I ` P ) .(+) ( I ` Q ) ) = ( ( I ` Q ) .(+) ( I ` P ) ) ) |
60 |
39 44 59
|
3eqtr4d |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ Q ( le ` K ) W ) ) -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) ) |
61 |
7
|
adantr |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> ( K e. HL /\ W e. H ) ) |
62 |
8
|
adantr |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> P e. A ) |
63 |
|
simprl |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> -. P ( le ` K ) W ) |
64 |
62 63
|
jca |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> ( P e. A /\ -. P ( le ` K ) W ) ) |
65 |
9
|
adantr |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> Q e. A ) |
66 |
|
simprr |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> -. Q ( le ` K ) W ) |
67 |
65 66
|
jca |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> ( Q e. A /\ -. Q ( le ` K ) W ) ) |
68 |
10 1 2 3 4 5 6 61 64 67
|
dihjatcc |
|- ( ( ph /\ ( -. P ( le ` K ) W /\ -. Q ( le ` K ) W ) ) -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) ) |
69 |
18 29 60 68
|
4casesdan |
|- ( ph -> ( I ` ( P .\/ Q ) ) = ( ( I ` P ) .(+) ( I ` Q ) ) ) |