Metamath Proof Explorer

Theorem lsmdisj2b

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 )
Assertion lsmdisj2b 
|- ( ph -> ( ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) <-> ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( 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 incom 
 |-  ( S i^i ( T .(+) U ) ) = ( ( T .(+) U ) i^i S )
7 3 adantr 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> T e. ( SubGrp ` G ) )
8 2 adantr 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> S e. ( SubGrp ` G ) )
9 4 adantr 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> U e. ( SubGrp ` G ) )
10 incom 
 |-  ( T i^i ( S .(+) U ) ) = ( ( S .(+) U ) i^i T )
11 simprl 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( S .(+) U ) i^i T ) = { .0. } )
12 10 11 eqtrid 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( T i^i ( S .(+) U ) ) = { .0. } )
13 simprr 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( S i^i U ) = { .0. } )
14 1 7 8 9 5 12 13 lsmdisj2r 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( T .(+) U ) i^i S ) = { .0. } )
15 6 14 eqtrid 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( S i^i ( T .(+) U ) ) = { .0. } )
16 incom 
 |-  ( T i^i U ) = ( U i^i T )
17 1 8 9 7 5 11 lsmdisj 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( S i^i T ) = { .0. } /\ ( U i^i T ) = { .0. } ) )
18 17 simprd 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( U i^i T ) = { .0. } )
19 16 18 eqtrid 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( T i^i U ) = { .0. } )
20 15 19 jca 
 |-  ( ( ph /\ ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) )
21 2 adantr 
 |-  ( ( ph /\ ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) -> S e. ( SubGrp ` G ) )
22 3 adantr 
 |-  ( ( ph /\ ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) -> T e. ( SubGrp ` G ) )
23 4 adantr 
 |-  ( ( ph /\ ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) -> U e. ( SubGrp ` G ) )
24 simprl 
 |-  ( ( ph /\ ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) -> ( S i^i ( T .(+) U ) ) = { .0. } )
25 simprr 
 |-  ( ( ph /\ ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) -> ( T i^i U ) = { .0. } )
26 1 21 22 23 5 24 25 lsmdisj2r 
 |-  ( ( ph /\ ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) -> ( ( S .(+) U ) i^i T ) = { .0. } )
27 1 21 22 23 5 24 lsmdisjr 
 |-  ( ( ph /\ ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) -> ( ( S i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) )
28 27 simprd 
 |-  ( ( ph /\ ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) -> ( S i^i U ) = { .0. } )
29 26 28 jca 
 |-  ( ( ph /\ ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) -> ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) )
30 20 29 impbida 
 |-  ( ph -> ( ( ( ( S .(+) U ) i^i T ) = { .0. } /\ ( S i^i U ) = { .0. } ) <-> ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) )