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 |
|
dmdprdsplit.z |
|- Z = ( Cntz ` G ) |
5 |
|
dmdprdsplit.0 |
|- .0. = ( 0g ` G ) |
6 |
|
dmdprdsplit2.1 |
|- ( ph -> G dom DProd ( S |` C ) ) |
7 |
|
dmdprdsplit2.2 |
|- ( ph -> G dom DProd ( S |` D ) ) |
8 |
|
dmdprdsplit2.3 |
|- ( ph -> ( G DProd ( S |` C ) ) C_ ( Z ` ( G DProd ( S |` D ) ) ) ) |
9 |
|
dmdprdsplit2.4 |
|- ( ph -> ( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = { .0. } ) |
10 |
|
eqid |
|- ( mrCls ` ( SubGrp ` G ) ) = ( mrCls ` ( SubGrp ` G ) ) |
11 |
|
dprdgrp |
|- ( G dom DProd ( S |` C ) -> G e. Grp ) |
12 |
6 11
|
syl |
|- ( ph -> G e. Grp ) |
13 |
|
ssun1 |
|- C C_ ( C u. D ) |
14 |
13 3
|
sseqtrrid |
|- ( ph -> C C_ I ) |
15 |
1 14
|
fssresd |
|- ( ph -> ( S |` C ) : C --> ( SubGrp ` G ) ) |
16 |
15
|
fdmd |
|- ( ph -> dom ( S |` C ) = C ) |
17 |
6 16
|
dprddomcld |
|- ( ph -> C e. _V ) |
18 |
|
ssun2 |
|- D C_ ( C u. D ) |
19 |
18 3
|
sseqtrrid |
|- ( ph -> D C_ I ) |
20 |
1 19
|
fssresd |
|- ( ph -> ( S |` D ) : D --> ( SubGrp ` G ) ) |
21 |
20
|
fdmd |
|- ( ph -> dom ( S |` D ) = D ) |
22 |
7 21
|
dprddomcld |
|- ( ph -> D e. _V ) |
23 |
|
unexg |
|- ( ( C e. _V /\ D e. _V ) -> ( C u. D ) e. _V ) |
24 |
17 22 23
|
syl2anc |
|- ( ph -> ( C u. D ) e. _V ) |
25 |
3 24
|
eqeltrd |
|- ( ph -> I e. _V ) |
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 |
1 2 3 4 5 6 7 8 9 10
|
dmdprdsplit2lem |
|- ( ( ph /\ x e. C ) -> ( ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) /\ ( ( S ` x ) i^i ( ( mrCls ` ( SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } ) ) |
30 |
|
incom |
|- ( C i^i D ) = ( D i^i C ) |
31 |
30 2
|
eqtr3id |
|- ( ph -> ( D i^i C ) = (/) ) |
32 |
|
uncom |
|- ( C u. D ) = ( D u. C ) |
33 |
3 32
|
eqtrdi |
|- ( ph -> I = ( D u. C ) ) |
34 |
|
dprdsubg |
|- ( G dom DProd ( S |` C ) -> ( G DProd ( S |` C ) ) e. ( SubGrp ` G ) ) |
35 |
6 34
|
syl |
|- ( ph -> ( G DProd ( S |` C ) ) e. ( SubGrp ` G ) ) |
36 |
|
dprdsubg |
|- ( G dom DProd ( S |` D ) -> ( G DProd ( S |` D ) ) e. ( SubGrp ` G ) ) |
37 |
7 36
|
syl |
|- ( ph -> ( G DProd ( S |` D ) ) e. ( SubGrp ` G ) ) |
38 |
4 35 37 8
|
cntzrecd |
|- ( ph -> ( G DProd ( S |` D ) ) C_ ( Z ` ( G DProd ( S |` C ) ) ) ) |
39 |
|
incom |
|- ( ( G DProd ( S |` C ) ) i^i ( G DProd ( S |` D ) ) ) = ( ( G DProd ( S |` D ) ) i^i ( G DProd ( S |` C ) ) ) |
40 |
39 9
|
eqtr3id |
|- ( ph -> ( ( G DProd ( S |` D ) ) i^i ( G DProd ( S |` C ) ) ) = { .0. } ) |
41 |
1 31 33 4 5 7 6 38 40 10
|
dmdprdsplit2lem |
|- ( ( ph /\ x e. D ) -> ( ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) /\ ( ( S ` x ) i^i ( ( mrCls ` ( SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } ) ) |
42 |
29 41
|
jaodan |
|- ( ( ph /\ ( x e. C \/ x e. D ) ) -> ( ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) /\ ( ( S ` x ) i^i ( ( mrCls ` ( SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } ) ) |
43 |
42
|
simpld |
|- ( ( ph /\ ( x e. C \/ x e. D ) ) -> ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) ) |
44 |
43
|
ex |
|- ( ph -> ( ( x e. C \/ x e. D ) -> ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) ) ) |
45 |
28 44
|
sylbid |
|- ( ph -> ( x e. I -> ( y e. I -> ( x =/= y -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) ) ) ) |
46 |
45
|
3imp2 |
|- ( ( ph /\ ( x e. I /\ y e. I /\ x =/= y ) ) -> ( S ` x ) C_ ( Z ` ( S ` y ) ) ) |
47 |
28
|
biimpa |
|- ( ( ph /\ x e. I ) -> ( x e. C \/ x e. D ) ) |
48 |
29
|
simprd |
|- ( ( ph /\ x e. C ) -> ( ( S ` x ) i^i ( ( mrCls ` ( SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } ) |
49 |
41
|
simprd |
|- ( ( ph /\ x e. D ) -> ( ( S ` x ) i^i ( ( mrCls ` ( SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } ) |
50 |
48 49
|
jaodan |
|- ( ( ph /\ ( x e. C \/ x e. D ) ) -> ( ( S ` x ) i^i ( ( mrCls ` ( SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } ) |
51 |
47 50
|
syldan |
|- ( ( ph /\ x e. I ) -> ( ( S ` x ) i^i ( ( mrCls ` ( SubGrp ` G ) ) ` U. ( S " ( I \ { x } ) ) ) ) C_ { .0. } ) |
52 |
4 5 10 12 25 1 46 51
|
dmdprdd |
|- ( ph -> G dom DProd S ) |