| 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 |
|
subgrcl |
|- ( T e. ( SubGrp ` G ) -> G e. Grp ) |
| 11 |
5 10
|
syl |
|- ( ph -> G e. Grp ) |
| 12 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
| 13 |
12
|
subgss |
|- ( T e. ( SubGrp ` G ) -> T C_ ( Base ` G ) ) |
| 14 |
5 13
|
syl |
|- ( ph -> T C_ ( Base ` G ) ) |
| 15 |
12
|
subgss |
|- ( U e. ( SubGrp ` G ) -> U C_ ( Base ` G ) ) |
| 16 |
6 15
|
syl |
|- ( ph -> U C_ ( Base ` G ) ) |
| 17 |
12 1 2 9
|
pj1fval |
|- ( ( G e. Grp /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) -> ( T P U ) = ( z e. ( T .(+) U ) |-> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) ) ) |
| 18 |
11 14 16 17
|
syl3anc |
|- ( ph -> ( T P U ) = ( z e. ( T .(+) U ) |-> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) ) ) |
| 19 |
1 2 3 4 5 6 7 8
|
pj1eu |
|- ( ( ph /\ z e. ( T .(+) U ) ) -> E! x e. T E. y e. U z = ( x .+ y ) ) |
| 20 |
|
riotacl |
|- ( E! x e. T E. y e. U z = ( x .+ y ) -> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) e. T ) |
| 21 |
19 20
|
syl |
|- ( ( ph /\ z e. ( T .(+) U ) ) -> ( iota_ x e. T E. y e. U z = ( x .+ y ) ) e. T ) |
| 22 |
18 21
|
fmpt3d |
|- ( ph -> ( T P U ) : ( T .(+) U ) --> T ) |