Step |
Hyp |
Ref |
Expression |
1 |
|
lsmcom.s |
|- .(+) = ( LSSum ` G ) |
2 |
|
simp1 |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> G e. Abel ) |
3 |
|
simp2r |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> R e. ( SubGrp ` G ) ) |
4 |
|
simp3l |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> T e. ( SubGrp ` G ) ) |
5 |
1
|
lsmcom |
|- ( ( G e. Abel /\ R e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) ) -> ( R .(+) T ) = ( T .(+) R ) ) |
6 |
2 3 4 5
|
syl3anc |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( R .(+) T ) = ( T .(+) R ) ) |
7 |
6
|
oveq2d |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( Q .(+) ( R .(+) T ) ) = ( Q .(+) ( T .(+) R ) ) ) |
8 |
|
simp2l |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> Q e. ( SubGrp ` G ) ) |
9 |
1
|
lsmass |
|- ( ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) ) -> ( ( Q .(+) R ) .(+) T ) = ( Q .(+) ( R .(+) T ) ) ) |
10 |
8 3 4 9
|
syl3anc |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( ( Q .(+) R ) .(+) T ) = ( Q .(+) ( R .(+) T ) ) ) |
11 |
1
|
lsmass |
|- ( ( Q e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) -> ( ( Q .(+) T ) .(+) R ) = ( Q .(+) ( T .(+) R ) ) ) |
12 |
8 4 3 11
|
syl3anc |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( ( Q .(+) T ) .(+) R ) = ( Q .(+) ( T .(+) R ) ) ) |
13 |
7 10 12
|
3eqtr4d |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( ( Q .(+) R ) .(+) T ) = ( ( Q .(+) T ) .(+) R ) ) |
14 |
13
|
oveq1d |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( ( ( Q .(+) R ) .(+) T ) .(+) U ) = ( ( ( Q .(+) T ) .(+) R ) .(+) U ) ) |
15 |
1
|
lsmsubg2 |
|- ( ( G e. Abel /\ Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) -> ( Q .(+) R ) e. ( SubGrp ` G ) ) |
16 |
2 8 3 15
|
syl3anc |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( Q .(+) R ) e. ( SubGrp ` G ) ) |
17 |
|
simp3r |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> U e. ( SubGrp ` G ) ) |
18 |
1
|
lsmass |
|- ( ( ( Q .(+) R ) e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( ( ( Q .(+) R ) .(+) T ) .(+) U ) = ( ( Q .(+) R ) .(+) ( T .(+) U ) ) ) |
19 |
16 4 17 18
|
syl3anc |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( ( ( Q .(+) R ) .(+) T ) .(+) U ) = ( ( Q .(+) R ) .(+) ( T .(+) U ) ) ) |
20 |
1
|
lsmsubg2 |
|- ( ( G e. Abel /\ Q e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) ) -> ( Q .(+) T ) e. ( SubGrp ` G ) ) |
21 |
2 8 4 20
|
syl3anc |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( Q .(+) T ) e. ( SubGrp ` G ) ) |
22 |
1
|
lsmass |
|- ( ( ( Q .(+) T ) e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) -> ( ( ( Q .(+) T ) .(+) R ) .(+) U ) = ( ( Q .(+) T ) .(+) ( R .(+) U ) ) ) |
23 |
21 3 17 22
|
syl3anc |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( ( ( Q .(+) T ) .(+) R ) .(+) U ) = ( ( Q .(+) T ) .(+) ( R .(+) U ) ) ) |
24 |
14 19 23
|
3eqtr3d |
|- ( ( G e. Abel /\ ( Q e. ( SubGrp ` G ) /\ R e. ( SubGrp ` G ) ) /\ ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) ) ) -> ( ( Q .(+) R ) .(+) ( T .(+) U ) ) = ( ( Q .(+) T ) .(+) ( R .(+) U ) ) ) |