Step |
Hyp |
Ref |
Expression |
1 |
|
dmdprdpr.z |
|- Z = ( Cntz ` G ) |
2 |
|
dmdprdpr.0 |
|- .0. = ( 0g ` G ) |
3 |
|
dmdprdpr.s |
|- ( ph -> S e. ( SubGrp ` G ) ) |
4 |
|
dmdprdpr.t |
|- ( ph -> T e. ( SubGrp ` G ) ) |
5 |
|
0ex |
|- (/) e. _V |
6 |
|
dprdsn |
|- ( ( (/) e. _V /\ S e. ( SubGrp ` G ) ) -> ( G dom DProd { <. (/) , S >. } /\ ( G DProd { <. (/) , S >. } ) = S ) ) |
7 |
5 3 6
|
sylancr |
|- ( ph -> ( G dom DProd { <. (/) , S >. } /\ ( G DProd { <. (/) , S >. } ) = S ) ) |
8 |
7
|
simpld |
|- ( ph -> G dom DProd { <. (/) , S >. } ) |
9 |
|
xpscf |
|- ( { <. (/) , S >. , <. 1o , T >. } : 2o --> ( SubGrp ` G ) <-> ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) ) ) |
10 |
3 4 9
|
sylanbrc |
|- ( ph -> { <. (/) , S >. , <. 1o , T >. } : 2o --> ( SubGrp ` G ) ) |
11 |
10
|
ffnd |
|- ( ph -> { <. (/) , S >. , <. 1o , T >. } Fn 2o ) |
12 |
5
|
prid1 |
|- (/) e. { (/) , 1o } |
13 |
|
df2o3 |
|- 2o = { (/) , 1o } |
14 |
12 13
|
eleqtrri |
|- (/) e. 2o |
15 |
|
fnressn |
|- ( ( { <. (/) , S >. , <. 1o , T >. } Fn 2o /\ (/) e. 2o ) -> ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) = { <. (/) , ( { <. (/) , S >. , <. 1o , T >. } ` (/) ) >. } ) |
16 |
11 14 15
|
sylancl |
|- ( ph -> ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) = { <. (/) , ( { <. (/) , S >. , <. 1o , T >. } ` (/) ) >. } ) |
17 |
|
fvpr0o |
|- ( S e. ( SubGrp ` G ) -> ( { <. (/) , S >. , <. 1o , T >. } ` (/) ) = S ) |
18 |
3 17
|
syl |
|- ( ph -> ( { <. (/) , S >. , <. 1o , T >. } ` (/) ) = S ) |
19 |
18
|
opeq2d |
|- ( ph -> <. (/) , ( { <. (/) , S >. , <. 1o , T >. } ` (/) ) >. = <. (/) , S >. ) |
20 |
19
|
sneqd |
|- ( ph -> { <. (/) , ( { <. (/) , S >. , <. 1o , T >. } ` (/) ) >. } = { <. (/) , S >. } ) |
21 |
16 20
|
eqtrd |
|- ( ph -> ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) = { <. (/) , S >. } ) |
22 |
8 21
|
breqtrrd |
|- ( ph -> G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) |
23 |
|
1on |
|- 1o e. On |
24 |
|
dprdsn |
|- ( ( 1o e. On /\ T e. ( SubGrp ` G ) ) -> ( G dom DProd { <. 1o , T >. } /\ ( G DProd { <. 1o , T >. } ) = T ) ) |
25 |
23 4 24
|
sylancr |
|- ( ph -> ( G dom DProd { <. 1o , T >. } /\ ( G DProd { <. 1o , T >. } ) = T ) ) |
26 |
25
|
simpld |
|- ( ph -> G dom DProd { <. 1o , T >. } ) |
27 |
|
1oex |
|- 1o e. _V |
28 |
27
|
prid2 |
|- 1o e. { (/) , 1o } |
29 |
28 13
|
eleqtrri |
|- 1o e. 2o |
30 |
|
fnressn |
|- ( ( { <. (/) , S >. , <. 1o , T >. } Fn 2o /\ 1o e. 2o ) -> ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) = { <. 1o , ( { <. (/) , S >. , <. 1o , T >. } ` 1o ) >. } ) |
31 |
11 29 30
|
sylancl |
|- ( ph -> ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) = { <. 1o , ( { <. (/) , S >. , <. 1o , T >. } ` 1o ) >. } ) |
32 |
|
fvpr1o |
|- ( T e. ( SubGrp ` G ) -> ( { <. (/) , S >. , <. 1o , T >. } ` 1o ) = T ) |
33 |
4 32
|
syl |
|- ( ph -> ( { <. (/) , S >. , <. 1o , T >. } ` 1o ) = T ) |
34 |
33
|
opeq2d |
|- ( ph -> <. 1o , ( { <. (/) , S >. , <. 1o , T >. } ` 1o ) >. = <. 1o , T >. ) |
35 |
34
|
sneqd |
|- ( ph -> { <. 1o , ( { <. (/) , S >. , <. 1o , T >. } ` 1o ) >. } = { <. 1o , T >. } ) |
36 |
31 35
|
eqtrd |
|- ( ph -> ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) = { <. 1o , T >. } ) |
37 |
26 36
|
breqtrrd |
|- ( ph -> G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) |
38 |
|
1n0 |
|- 1o =/= (/) |
39 |
38
|
necomi |
|- (/) =/= 1o |
40 |
|
disjsn2 |
|- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) ) |
41 |
39 40
|
mp1i |
|- ( ph -> ( { (/) } i^i { 1o } ) = (/) ) |
42 |
|
df-pr |
|- { (/) , 1o } = ( { (/) } u. { 1o } ) |
43 |
13 42
|
eqtri |
|- 2o = ( { (/) } u. { 1o } ) |
44 |
43
|
a1i |
|- ( ph -> 2o = ( { (/) } u. { 1o } ) ) |
45 |
10 41 44 1 2
|
dmdprdsplit |
|- ( ph -> ( G dom DProd { <. (/) , S >. , <. 1o , T >. } <-> ( ( G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) /\ G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) /\ ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) C_ ( Z ` ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) /\ ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) i^i ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = { .0. } ) ) ) |
46 |
|
3anass |
|- ( ( ( G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) /\ G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) /\ ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) C_ ( Z ` ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) /\ ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) i^i ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = { .0. } ) <-> ( ( G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) /\ G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) /\ ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) C_ ( Z ` ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) /\ ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) i^i ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = { .0. } ) ) ) |
47 |
45 46
|
bitrdi |
|- ( ph -> ( G dom DProd { <. (/) , S >. , <. 1o , T >. } <-> ( ( G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) /\ G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) /\ ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) C_ ( Z ` ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) /\ ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) i^i ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = { .0. } ) ) ) ) |
48 |
47
|
baibd |
|- ( ( ph /\ ( G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) /\ G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) -> ( G dom DProd { <. (/) , S >. , <. 1o , T >. } <-> ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) C_ ( Z ` ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) /\ ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) i^i ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = { .0. } ) ) ) |
49 |
48
|
ex |
|- ( ph -> ( ( G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) /\ G dom DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) -> ( G dom DProd { <. (/) , S >. , <. 1o , T >. } <-> ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) C_ ( Z ` ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) /\ ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) i^i ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = { .0. } ) ) ) ) |
50 |
22 37 49
|
mp2and |
|- ( ph -> ( G dom DProd { <. (/) , S >. , <. 1o , T >. } <-> ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) C_ ( Z ` ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) /\ ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) i^i ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = { .0. } ) ) ) |
51 |
21
|
oveq2d |
|- ( ph -> ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) = ( G DProd { <. (/) , S >. } ) ) |
52 |
7
|
simprd |
|- ( ph -> ( G DProd { <. (/) , S >. } ) = S ) |
53 |
51 52
|
eqtrd |
|- ( ph -> ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) = S ) |
54 |
36
|
oveq2d |
|- ( ph -> ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) = ( G DProd { <. 1o , T >. } ) ) |
55 |
25
|
simprd |
|- ( ph -> ( G DProd { <. 1o , T >. } ) = T ) |
56 |
54 55
|
eqtrd |
|- ( ph -> ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) = T ) |
57 |
56
|
fveq2d |
|- ( ph -> ( Z ` ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = ( Z ` T ) ) |
58 |
53 57
|
sseq12d |
|- ( ph -> ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) C_ ( Z ` ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) <-> S C_ ( Z ` T ) ) ) |
59 |
53 56
|
ineq12d |
|- ( ph -> ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) i^i ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = ( S i^i T ) ) |
60 |
59
|
eqeq1d |
|- ( ph -> ( ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) i^i ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = { .0. } <-> ( S i^i T ) = { .0. } ) ) |
61 |
58 60
|
anbi12d |
|- ( ph -> ( ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) C_ ( Z ` ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) /\ ( ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { (/) } ) ) i^i ( G DProd ( { <. (/) , S >. , <. 1o , T >. } |` { 1o } ) ) ) = { .0. } ) <-> ( S C_ ( Z ` T ) /\ ( S i^i T ) = { .0. } ) ) ) |
62 |
50 61
|
bitrd |
|- ( ph -> ( G dom DProd { <. (/) , S >. , <. 1o , T >. } <-> ( S C_ ( Z ` T ) /\ ( S i^i T ) = { .0. } ) ) ) |