Metamath Proof Explorer

Theorem lsmdisj2r

Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016)

Ref Expression
Hypotheses lsmcntz.p 
|- .(+) = ( LSSum ` G )
lsmcntz.s 
|- ( ph -> S e. ( SubGrp ` G ) )
lsmcntz.t 
|- ( ph -> T e. ( SubGrp ` G ) )
lsmcntz.u 
|- ( ph -> U e. ( SubGrp ` G ) )
lsmdisj.o 
|- .0. = ( 0g ` G )
lsmdisjr.i 
|- ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } )
lsmdisj2r.i 
|- ( ph -> ( T i^i U ) = { .0. } )
Assertion lsmdisj2r 
|- ( ph -> ( ( S .(+) U ) i^i T ) = { .0. } )

Proof

Step Hyp Ref Expression
1 lsmcntz.p 
 |-  .(+) = ( LSSum ` G )
2 lsmcntz.s 
 |-  ( ph -> S e. ( SubGrp ` G ) )
3 lsmcntz.t 
 |-  ( ph -> T e. ( SubGrp ` G ) )
4 lsmcntz.u 
 |-  ( ph -> U e. ( SubGrp ` G ) )
5 lsmdisj.o 
 |-  .0. = ( 0g ` G )
6 lsmdisjr.i 
 |-  ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } )
7 lsmdisj2r.i 
 |-  ( ph -> ( T i^i U ) = { .0. } )
8 eqid 
 |-  ( oppG ` G ) = ( oppG ` G )
9 8 1 oppglsm 
 |-  ( U ( LSSum ` ( oppG ` G ) ) S ) = ( S .(+) U )
10 9 ineq2i 
 |-  ( T i^i ( U ( LSSum ` ( oppG ` G ) ) S ) ) = ( T i^i ( S .(+) U ) )
11 incom 
 |-  ( T i^i ( S .(+) U ) ) = ( ( S .(+) U ) i^i T )
12 10 11 eqtri 
 |-  ( T i^i ( U ( LSSum ` ( oppG ` G ) ) S ) ) = ( ( S .(+) U ) i^i T )
13 eqid 
 |-  ( LSSum ` ( oppG ` G ) ) = ( LSSum ` ( oppG ` G ) )
14 8 oppgsubg 
 |-  ( SubGrp ` G ) = ( SubGrp ` ( oppG ` G ) )
15 4 14 eleqtrdi 
 |-  ( ph -> U e. ( SubGrp ` ( oppG ` G ) ) )
16 3 14 eleqtrdi 
 |-  ( ph -> T e. ( SubGrp ` ( oppG ` G ) ) )
17 2 14 eleqtrdi 
 |-  ( ph -> S e. ( SubGrp ` ( oppG ` G ) ) )
18 8 5 oppgid 
 |-  .0. = ( 0g ` ( oppG ` G ) )
19 8 1 oppglsm 
 |-  ( U ( LSSum ` ( oppG ` G ) ) T ) = ( T .(+) U )
20 19 ineq1i 
 |-  ( ( U ( LSSum ` ( oppG ` G ) ) T ) i^i S ) = ( ( T .(+) U ) i^i S )
21 incom 
 |-  ( ( T .(+) U ) i^i S ) = ( S i^i ( T .(+) U ) )
22 20 21 eqtri 
 |-  ( ( U ( LSSum ` ( oppG ` G ) ) T ) i^i S ) = ( S i^i ( T .(+) U ) )
23 22 6 syl5eq 
 |-  ( ph -> ( ( U ( LSSum ` ( oppG ` G ) ) T ) i^i S ) = { .0. } )
24 incom 
 |-  ( T i^i U ) = ( U i^i T )
25 24 7 eqtr3id 
 |-  ( ph -> ( U i^i T ) = { .0. } )
26 13 15 16 17 18 23 25 lsmdisj2 
 |-  ( ph -> ( T i^i ( U ( LSSum ` ( oppG ` G ) ) S ) ) = { .0. } )
27 12 26 eqtr3id 
 |-  ( ph -> ( ( S .(+) U ) i^i T ) = { .0. } )