Step |
Hyp |
Ref |
Expression |
1 |
|
dprdsplit.2 |
|- ( ph -> S : I --> ( SubGrp ` G ) ) |
2 |
|
dprdsplit.i |
|- ( ph -> ( C i^i D ) = (/) ) |
3 |
|
dprdsplit.u |
|- ( ph -> I = ( C u. D ) ) |
4 |
|
dprdsplit.s |
|- .(+) = ( LSSum ` G ) |
5 |
|
dprdsplit.1 |
|- ( ph -> G dom DProd S ) |
6 |
1
|
fdmd |
|- ( ph -> dom S = I ) |
7 |
|
ssun1 |
|- C C_ ( C u. D ) |
8 |
7 3
|
sseqtrrid |
|- ( ph -> C C_ I ) |
9 |
5 6 8
|
dprdres |
|- ( ph -> ( G dom DProd ( S |` C ) /\ ( G DProd ( S |` C ) ) C_ ( G DProd S ) ) ) |
10 |
9
|
simpld |
|- ( ph -> G dom DProd ( S |` C ) ) |
11 |
|
dprdsubg |
|- ( G dom DProd ( S |` C ) -> ( G DProd ( S |` C ) ) e. ( SubGrp ` G ) ) |
12 |
10 11
|
syl |
|- ( ph -> ( G DProd ( S |` C ) ) e. ( SubGrp ` G ) ) |
13 |
|
ssun2 |
|- D C_ ( C u. D ) |
14 |
13 3
|
sseqtrrid |
|- ( ph -> D C_ I ) |
15 |
5 6 14
|
dprdres |
|- ( ph -> ( G dom DProd ( S |` D ) /\ ( G DProd ( S |` D ) ) C_ ( G DProd S ) ) ) |
16 |
15
|
simpld |
|- ( ph -> G dom DProd ( S |` D ) ) |
17 |
|
dprdsubg |
|- ( G dom DProd ( S |` D ) -> ( G DProd ( S |` D ) ) e. ( SubGrp ` G ) ) |
18 |
16 17
|
syl |
|- ( ph -> ( G DProd ( S |` D ) ) e. ( SubGrp ` G ) ) |
19 |
|
eqid |
|- ( Cntz ` G ) = ( Cntz ` G ) |
20 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
21 |
1 2 3 19 20
|
dmdprdsplit |
|- ( ph -> ( G dom DProd S <-> ( ( G dom DProd ( S |` C ) /\ G dom DProd ( S |` D ) ) /\ ( G DProd ( S |` C ) ) C_ ( ( Cntz ` G ) ` ( G DProd ( S |` D ) ) ) /\ ( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = { ( 0g ` G ) } ) ) ) |
22 |
5 21
|
mpbid |
|- ( ph -> ( ( G dom DProd ( S |` C ) /\ G dom DProd ( S |` D ) ) /\ ( G DProd ( S |` C ) ) C_ ( ( Cntz ` G ) ` ( G DProd ( S |` D ) ) ) /\ ( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = { ( 0g ` G ) } ) ) |
23 |
22
|
simp2d |
|- ( ph -> ( G DProd ( S |` C ) ) C_ ( ( Cntz ` G ) ` ( G DProd ( S |` D ) ) ) ) |
24 |
4 19
|
lsmsubg |
|- ( ( ( G DProd ( S |` C ) ) e. ( SubGrp ` G ) /\ ( G DProd ( S |` D ) ) e. ( SubGrp ` G ) /\ ( G DProd ( S |` C ) ) C_ ( ( Cntz ` G ) ` ( G DProd ( S |` D ) ) ) ) -> ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) e. ( SubGrp ` G ) ) |
25 |
12 18 23 24
|
syl3anc |
|- ( ph -> ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) e. ( SubGrp ` G ) ) |
26 |
3
|
eleq2d |
|- ( ph -> ( x e. I <-> x e. ( C u. D ) ) ) |
27 |
|
elun |
|- ( x e. ( C u. D ) <-> ( x e. C \/ x e. D ) ) |
28 |
26 27
|
bitrdi |
|- ( ph -> ( x e. I <-> ( x e. C \/ x e. D ) ) ) |
29 |
28
|
biimpa |
|- ( ( ph /\ x e. I ) -> ( x e. C \/ x e. D ) ) |
30 |
|
fvres |
|- ( x e. C -> ( ( S |` C ) ` x ) = ( S ` x ) ) |
31 |
30
|
adantl |
|- ( ( ph /\ x e. C ) -> ( ( S |` C ) ` x ) = ( S ` x ) ) |
32 |
10
|
adantr |
|- ( ( ph /\ x e. C ) -> G dom DProd ( S |` C ) ) |
33 |
1 8
|
fssresd |
|- ( ph -> ( S |` C ) : C --> ( SubGrp ` G ) ) |
34 |
33
|
fdmd |
|- ( ph -> dom ( S |` C ) = C ) |
35 |
34
|
adantr |
|- ( ( ph /\ x e. C ) -> dom ( S |` C ) = C ) |
36 |
|
simpr |
|- ( ( ph /\ x e. C ) -> x e. C ) |
37 |
32 35 36
|
dprdub |
|- ( ( ph /\ x e. C ) -> ( ( S |` C ) ` x ) C_ ( G DProd ( S |` C ) ) ) |
38 |
31 37
|
eqsstrrd |
|- ( ( ph /\ x e. C ) -> ( S ` x ) C_ ( G DProd ( S |` C ) ) ) |
39 |
4
|
lsmub1 |
|- ( ( ( G DProd ( S |` C ) ) e. ( SubGrp ` G ) /\ ( G DProd ( S |` D ) ) e. ( SubGrp ` G ) ) -> ( G DProd ( S |` C ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
40 |
12 18 39
|
syl2anc |
|- ( ph -> ( G DProd ( S |` C ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
41 |
40
|
adantr |
|- ( ( ph /\ x e. C ) -> ( G DProd ( S |` C ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
42 |
38 41
|
sstrd |
|- ( ( ph /\ x e. C ) -> ( S ` x ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
43 |
|
fvres |
|- ( x e. D -> ( ( S |` D ) ` x ) = ( S ` x ) ) |
44 |
43
|
adantl |
|- ( ( ph /\ x e. D ) -> ( ( S |` D ) ` x ) = ( S ` x ) ) |
45 |
16
|
adantr |
|- ( ( ph /\ x e. D ) -> G dom DProd ( S |` D ) ) |
46 |
1 14
|
fssresd |
|- ( ph -> ( S |` D ) : D --> ( SubGrp ` G ) ) |
47 |
46
|
fdmd |
|- ( ph -> dom ( S |` D ) = D ) |
48 |
47
|
adantr |
|- ( ( ph /\ x e. D ) -> dom ( S |` D ) = D ) |
49 |
|
simpr |
|- ( ( ph /\ x e. D ) -> x e. D ) |
50 |
45 48 49
|
dprdub |
|- ( ( ph /\ x e. D ) -> ( ( S |` D ) ` x ) C_ ( G DProd ( S |` D ) ) ) |
51 |
44 50
|
eqsstrrd |
|- ( ( ph /\ x e. D ) -> ( S ` x ) C_ ( G DProd ( S |` D ) ) ) |
52 |
4
|
lsmub2 |
|- ( ( ( G DProd ( S |` C ) ) e. ( SubGrp ` G ) /\ ( G DProd ( S |` D ) ) e. ( SubGrp ` G ) ) -> ( G DProd ( S |` D ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
53 |
12 18 52
|
syl2anc |
|- ( ph -> ( G DProd ( S |` D ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
54 |
53
|
adantr |
|- ( ( ph /\ x e. D ) -> ( G DProd ( S |` D ) ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
55 |
51 54
|
sstrd |
|- ( ( ph /\ x e. D ) -> ( S ` x ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
56 |
42 55
|
jaodan |
|- ( ( ph /\ ( x e. C \/ x e. D ) ) -> ( S ` x ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
57 |
29 56
|
syldan |
|- ( ( ph /\ x e. I ) -> ( S ` x ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
58 |
5 6 25 57
|
dprdlub |
|- ( ph -> ( G DProd S ) C_ ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |
59 |
9
|
simprd |
|- ( ph -> ( G DProd ( S |` C ) ) C_ ( G DProd S ) ) |
60 |
15
|
simprd |
|- ( ph -> ( G DProd ( S |` D ) ) C_ ( G DProd S ) ) |
61 |
|
dprdsubg |
|- ( G dom DProd S -> ( G DProd S ) e. ( SubGrp ` G ) ) |
62 |
5 61
|
syl |
|- ( ph -> ( G DProd S ) e. ( SubGrp ` G ) ) |
63 |
4
|
lsmlub |
|- ( ( ( G DProd ( S |` C ) ) e. ( SubGrp ` G ) /\ ( G DProd ( S |` D ) ) e. ( SubGrp ` G ) /\ ( G DProd S ) e. ( SubGrp ` G ) ) -> ( ( ( G DProd ( S |` C ) ) C_ ( G DProd S ) /\ ( G DProd ( S |` D ) ) C_ ( G DProd S ) ) <-> ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) C_ ( G DProd S ) ) ) |
64 |
12 18 62 63
|
syl3anc |
|- ( ph -> ( ( ( G DProd ( S |` C ) ) C_ ( G DProd S ) /\ ( G DProd ( S |` D ) ) C_ ( G DProd S ) ) <-> ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) C_ ( G DProd S ) ) ) |
65 |
59 60 64
|
mpbi2and |
|- ( ph -> ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) C_ ( G DProd S ) ) |
66 |
58 65
|
eqssd |
|- ( ph -> ( G DProd S ) = ( ( G DProd ( S |` C ) ) .(+) ( G DProd ( S |` D ) ) ) ) |