This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lsmdisj3

Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-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 )
lsmdisj.i 
|- ( ph -> ( ( S .(+) T ) i^i U ) = { .0. } )
lsmdisj2.i 
|- ( ph -> ( S i^i T ) = { .0. } )
lsmdisj3.z 
|- Z = ( Cntz ` G )
lsmdisj3.s 
|- ( ph -> S C_ ( Z ` T ) )
Assertion lsmdisj3 
|- ( ph -> ( S i^i ( T .(+) 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 lsmdisj.i 
 |-  ( ph -> ( ( S .(+) T ) i^i U ) = { .0. } )
7 lsmdisj2.i 
 |-  ( ph -> ( S i^i T ) = { .0. } )
8 lsmdisj3.z 
 |-  Z = ( Cntz ` G )
9 lsmdisj3.s 
 |-  ( ph -> S C_ ( Z ` T ) )
10 1 8 lsmcom2 
 |-  ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) /\ S C_ ( Z ` T ) ) -> ( S .(+) T ) = ( T .(+) S ) )
11 2 3 9 10 syl3anc 
 |-  ( ph -> ( S .(+) T ) = ( T .(+) S ) )
12 11 ineq1d 
 |-  ( ph -> ( ( S .(+) T ) i^i U ) = ( ( T .(+) S ) i^i U ) )
13 12 6 eqtr3d 
 |-  ( ph -> ( ( T .(+) S ) i^i U ) = { .0. } )
14 incom 
 |-  ( T i^i S ) = ( S i^i T )
15 14 7 eqtrid 
 |-  ( ph -> ( T i^i S ) = { .0. } )
16 1 3 2 4 5 13 15 lsmdisj2 
 |-  ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } )