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 |
5
|
adantr |
|- ( ( ph /\ X e. U ) -> T e. ( SubGrp ` G ) ) |
11 |
|
subgrcl |
|- ( T e. ( SubGrp ` G ) -> G e. Grp ) |
12 |
10 11
|
syl |
|- ( ( ph /\ X e. U ) -> G e. Grp ) |
13 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
14 |
13
|
subgss |
|- ( U e. ( SubGrp ` G ) -> U C_ ( Base ` G ) ) |
15 |
6 14
|
syl |
|- ( ph -> U C_ ( Base ` G ) ) |
16 |
15
|
sselda |
|- ( ( ph /\ X e. U ) -> X e. ( Base ` G ) ) |
17 |
13 1 3
|
grplid |
|- ( ( G e. Grp /\ X e. ( Base ` G ) ) -> ( .0. .+ X ) = X ) |
18 |
12 16 17
|
syl2anc |
|- ( ( ph /\ X e. U ) -> ( .0. .+ X ) = X ) |
19 |
18
|
eqcomd |
|- ( ( ph /\ X e. U ) -> X = ( .0. .+ X ) ) |
20 |
6
|
adantr |
|- ( ( ph /\ X e. U ) -> U e. ( SubGrp ` G ) ) |
21 |
7
|
adantr |
|- ( ( ph /\ X e. U ) -> ( T i^i U ) = { .0. } ) |
22 |
8
|
adantr |
|- ( ( ph /\ X e. U ) -> T C_ ( Z ` U ) ) |
23 |
2
|
lsmub2 |
|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> U C_ ( T .(+) U ) ) |
24 |
5 6 23
|
syl2anc |
|- ( ph -> U C_ ( T .(+) U ) ) |
25 |
24
|
sselda |
|- ( ( ph /\ X e. U ) -> X e. ( T .(+) U ) ) |
26 |
3
|
subg0cl |
|- ( T e. ( SubGrp ` G ) -> .0. e. T ) |
27 |
10 26
|
syl |
|- ( ( ph /\ X e. U ) -> .0. e. T ) |
28 |
|
simpr |
|- ( ( ph /\ X e. U ) -> X e. U ) |
29 |
1 2 3 4 10 20 21 22 9 25 27 28
|
pj1eq |
|- ( ( ph /\ X e. U ) -> ( X = ( .0. .+ X ) <-> ( ( ( T P U ) ` X ) = .0. /\ ( ( U P T ) ` X ) = X ) ) ) |
30 |
19 29
|
mpbid |
|- ( ( ph /\ X e. U ) -> ( ( ( T P U ) ` X ) = .0. /\ ( ( U P T ) ` X ) = X ) ) |
31 |
30
|
simpld |
|- ( ( ph /\ X e. U ) -> ( ( T P U ) ` X ) = .0. ) |