Step |
Hyp |
Ref |
Expression |
1 |
|
lsmless2.v |
|- B = ( Base ` G ) |
2 |
|
lsmless2.s |
|- .(+) = ( LSSum ` G ) |
3 |
|
submrcl |
|- ( T e. ( SubMnd ` G ) -> G e. Mnd ) |
4 |
3
|
ad2antrr |
|- ( ( ( T e. ( SubMnd ` G ) /\ U C_ B ) /\ x e. U ) -> G e. Mnd ) |
5 |
|
simpr |
|- ( ( T e. ( SubMnd ` G ) /\ U C_ B ) -> U C_ B ) |
6 |
5
|
sselda |
|- ( ( ( T e. ( SubMnd ` G ) /\ U C_ B ) /\ x e. U ) -> x e. B ) |
7 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
8 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
9 |
1 7 8
|
mndlid |
|- ( ( G e. Mnd /\ x e. B ) -> ( ( 0g ` G ) ( +g ` G ) x ) = x ) |
10 |
4 6 9
|
syl2anc |
|- ( ( ( T e. ( SubMnd ` G ) /\ U C_ B ) /\ x e. U ) -> ( ( 0g ` G ) ( +g ` G ) x ) = x ) |
11 |
1
|
submss |
|- ( T e. ( SubMnd ` G ) -> T C_ B ) |
12 |
11
|
ad2antrr |
|- ( ( ( T e. ( SubMnd ` G ) /\ U C_ B ) /\ x e. U ) -> T C_ B ) |
13 |
|
simplr |
|- ( ( ( T e. ( SubMnd ` G ) /\ U C_ B ) /\ x e. U ) -> U C_ B ) |
14 |
8
|
subm0cl |
|- ( T e. ( SubMnd ` G ) -> ( 0g ` G ) e. T ) |
15 |
14
|
ad2antrr |
|- ( ( ( T e. ( SubMnd ` G ) /\ U C_ B ) /\ x e. U ) -> ( 0g ` G ) e. T ) |
16 |
|
simpr |
|- ( ( ( T e. ( SubMnd ` G ) /\ U C_ B ) /\ x e. U ) -> x e. U ) |
17 |
1 7 2
|
lsmelvalix |
|- ( ( ( G e. Mnd /\ T C_ B /\ U C_ B ) /\ ( ( 0g ` G ) e. T /\ x e. U ) ) -> ( ( 0g ` G ) ( +g ` G ) x ) e. ( T .(+) U ) ) |
18 |
4 12 13 15 16 17
|
syl32anc |
|- ( ( ( T e. ( SubMnd ` G ) /\ U C_ B ) /\ x e. U ) -> ( ( 0g ` G ) ( +g ` G ) x ) e. ( T .(+) U ) ) |
19 |
10 18
|
eqeltrrd |
|- ( ( ( T e. ( SubMnd ` G ) /\ U C_ B ) /\ x e. U ) -> x e. ( T .(+) U ) ) |
20 |
19
|
ex |
|- ( ( T e. ( SubMnd ` G ) /\ U C_ B ) -> ( x e. U -> x e. ( T .(+) U ) ) ) |
21 |
20
|
ssrdv |
|- ( ( T e. ( SubMnd ` G ) /\ U C_ B ) -> U C_ ( T .(+) U ) ) |