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Metamath Proof Explorer

Theorem lsmdisj3r

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. } )
lsmdisj3r.z 
|- Z = ( Cntz ` G )
lsmdisj3r.s 
|- ( ph -> T C_ ( Z ` U ) )
Assertion lsmdisj3r 
|- ( ph -> ( ( S .(+) T ) i^i U ) = { .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 lsmdisj3r.z 
 |-  Z = ( Cntz ` G )
9 lsmdisj3r.s 
 |-  ( ph -> T C_ ( Z ` U ) )
10 1 8 lsmcom2 
 |-  ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) /\ T C_ ( Z ` U ) ) -> ( T .(+) U ) = ( U .(+) T ) )
11 3 4 9 10 syl3anc 
 |-  ( ph -> ( T .(+) U ) = ( U .(+) T ) )
12 11 ineq2d 
 |-  ( ph -> ( S i^i ( T .(+) U ) ) = ( S i^i ( U .(+) T ) ) )
13 12 6 eqtr3d 
 |-  ( ph -> ( S i^i ( U .(+) T ) ) = { .0. } )
14 incom 
 |-  ( U i^i T ) = ( T i^i U )
15 14 7 eqtrid 
 |-  ( ph -> ( U i^i T ) = { .0. } )
16 1 2 4 3 5 13 15 lsmdisj2r 
 |-  ( ph -> ( ( S .(+) T ) i^i U ) = { .0. } )