Metamath Proof Explorer

Theorem lsmdisj2a

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 lsmdisj2a 
|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S 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 2 adantr 
 |-  ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> S e. ( SubGrp ` G ) )
7 3 adantr 
 |-  ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> T e. ( SubGrp ` G ) )
8 4 adantr 
 |-  ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> U e. ( SubGrp ` G ) )
9 simprl 
 |-  ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( ( S .(+) T ) i^i U ) = { .0. } )
10 simprr 
 |-  ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( S i^i T ) = { .0. } )
11 1 6 7 8 5 9 10 lsmdisj2 
 |-  ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( T i^i ( S .(+) U ) ) = { .0. } )
12 1 6 7 8 5 9 lsmdisj 
 |-  ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( ( S i^i U ) = { .0. } /\ ( T i^i U ) = { .0. } ) )
13 12 simpld 
 |-  ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( S i^i U ) = { .0. } )
14 11 13 jca 
 |-  ( ( ph /\ ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) ) -> ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) )
15 incom 
 |-  ( ( S .(+) T ) i^i U ) = ( U i^i ( S .(+) T ) )
16 2 adantr 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> S e. ( SubGrp ` G ) )
17 4 adantr 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> U e. ( SubGrp ` G ) )
18 3 adantr 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> T e. ( SubGrp ` G ) )
19 incom 
 |-  ( ( S .(+) U ) i^i T ) = ( T i^i ( S .(+) U ) )
20 simprl 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( T i^i ( S .(+) U ) ) = { .0. } )
21 19 20 syl5eq 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( S .(+) U ) i^i T ) = { .0. } )
22 simprr 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( S i^i U ) = { .0. } )
23 1 16 17 18 5 21 22 lsmdisj2 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( U i^i ( S .(+) T ) ) = { .0. } )
24 15 23 syl5eq 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( S .(+) T ) i^i U ) = { .0. } )
25 incom 
 |-  ( S i^i T ) = ( T i^i S )
26 1 18 16 17 5 20 lsmdisjr 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( T i^i S ) = { .0. } /\ ( T i^i U ) = { .0. } ) )
27 26 simpld 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( T i^i S ) = { .0. } )
28 25 27 syl5eq 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( S i^i T ) = { .0. } )
29 24 28 jca 
 |-  ( ( ph /\ ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) -> ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) )
30 14 29 impbida 
 |-  ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( T i^i ( S .(+) U ) ) = { .0. } /\ ( S i^i U ) = { .0. } ) ) )