Step |
Hyp |
Ref |
Expression |
1 |
|
pj1eu.a |
|- .+ = ( +g ` G ) |
2 |
|
pj1eu.s |
|- .(+) = ( LSSum ` G ) |
3 |
|
pj1eu.o |
|- .0. = ( 0g ` G ) |
4 |
|
pj1eu.z |
|- Z = ( Cntz ` G ) |
5 |
|
pj1eu.2 |
|- ( ph -> T e. ( SubGrp ` G ) ) |
6 |
|
pj1eu.3 |
|- ( ph -> U e. ( SubGrp ` G ) ) |
7 |
|
pj1eu.4 |
|- ( ph -> ( T i^i U ) = { .0. } ) |
8 |
|
pj1eu.5 |
|- ( ph -> T C_ ( Z ` U ) ) |
9 |
|
pj1f.p |
|- P = ( proj1 ` G ) |
10 |
|
incom |
|- ( U i^i T ) = ( T i^i U ) |
11 |
10 7
|
syl5eq |
|- ( ph -> ( U i^i T ) = { .0. } ) |
12 |
4 5 6 8
|
cntzrecd |
|- ( ph -> U C_ ( Z ` T ) ) |
13 |
1 2 3 4 6 5 11 12 9
|
pj1f |
|- ( ph -> ( U P T ) : ( U .(+) T ) --> U ) |
14 |
2 4
|
lsmcom2 |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) /\ T C_ ( Z ` U ) ) -> ( T .(+) U ) = ( U .(+) T ) ) |
15 |
5 6 8 14
|
syl3anc |
|- ( ph -> ( T .(+) U ) = ( U .(+) T ) ) |
16 |
15
|
feq2d |
|- ( ph -> ( ( U P T ) : ( T .(+) U ) --> U <-> ( U P T ) : ( U .(+) T ) --> U ) ) |
17 |
13 16
|
mpbird |
|- ( ph -> ( U P T ) : ( T .(+) U ) --> U ) |