Step |
Hyp |
Ref |
Expression |
1 |
|
dpjfval.1 |
|- ( ph -> G dom DProd S ) |
2 |
|
dpjfval.2 |
|- ( ph -> dom S = I ) |
3 |
|
dpjfval.p |
|- P = ( G dProj S ) |
4 |
|
dpjf.3 |
|- ( ph -> X e. I ) |
5 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
6 |
|
eqid |
|- ( LSSum ` G ) = ( LSSum ` G ) |
7 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
8 |
|
eqid |
|- ( Cntz ` G ) = ( Cntz ` G ) |
9 |
1 2
|
dprdf2 |
|- ( ph -> S : I --> ( SubGrp ` G ) ) |
10 |
9 4
|
ffvelrnd |
|- ( ph -> ( S ` X ) e. ( SubGrp ` G ) ) |
11 |
|
difssd |
|- ( ph -> ( I \ { X } ) C_ I ) |
12 |
1 2 11
|
dprdres |
|- ( ph -> ( G dom DProd ( S |` ( I \ { X } ) ) /\ ( G DProd ( S |` ( I \ { X } ) ) ) C_ ( G DProd S ) ) ) |
13 |
12
|
simpld |
|- ( ph -> G dom DProd ( S |` ( I \ { X } ) ) ) |
14 |
|
dprdsubg |
|- ( G dom DProd ( S |` ( I \ { X } ) ) -> ( G DProd ( S |` ( I \ { X } ) ) ) e. ( SubGrp ` G ) ) |
15 |
13 14
|
syl |
|- ( ph -> ( G DProd ( S |` ( I \ { X } ) ) ) e. ( SubGrp ` G ) ) |
16 |
1 2 4 7
|
dpjdisj |
|- ( ph -> ( ( S ` X ) i^i ( G DProd ( S |` ( I \ { X } ) ) ) ) = { ( 0g ` G ) } ) |
17 |
1 2 4 8
|
dpjcntz |
|- ( ph -> ( S ` X ) C_ ( ( Cntz ` G ) ` ( G DProd ( S |` ( I \ { X } ) ) ) ) ) |
18 |
|
eqid |
|- ( proj1 ` G ) = ( proj1 ` G ) |
19 |
5 6 7 8 10 15 16 17 18
|
pj1f |
|- ( ph -> ( ( S ` X ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { X } ) ) ) ) : ( ( S ` X ) ( LSSum ` G ) ( G DProd ( S |` ( I \ { X } ) ) ) ) --> ( S ` X ) ) |
20 |
1 2 3 18 4
|
dpjval |
|- ( ph -> ( P ` X ) = ( ( S ` X ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { X } ) ) ) ) ) |
21 |
1 2 4 6
|
dpjlsm |
|- ( ph -> ( G DProd S ) = ( ( S ` X ) ( LSSum ` G ) ( G DProd ( S |` ( I \ { X } ) ) ) ) ) |
22 |
20 21
|
feq12d |
|- ( ph -> ( ( P ` X ) : ( G DProd S ) --> ( S ` X ) <-> ( ( S ` X ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { X } ) ) ) ) : ( ( S ` X ) ( LSSum ` G ) ( G DProd ( S |` ( I \ { X } ) ) ) ) --> ( S ` X ) ) ) |
23 |
19 22
|
mpbird |
|- ( ph -> ( P ` X ) : ( G DProd S ) --> ( S ` X ) ) |