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Metamath Proof Explorer

Theorem lsmelvalmi

Description: Membership of vector subtraction in subgroup sum. (Contributed by NM, 27-Apr-2015) (Revised by Mario Carneiro, 19-Apr-2016)

Ref Expression
Hypotheses lsmelvalm.m 
|- .- = ( -g ` G )
lsmelvalm.p 
|- .(+) = ( LSSum ` G )
lsmelvalm.t 
|- ( ph -> T e. ( SubGrp ` G ) )
lsmelvalm.u 
|- ( ph -> U e. ( SubGrp ` G ) )
lsmelvalmi.x 
|- ( ph -> X e. T )
lsmelvalmi.y 
|- ( ph -> Y e. U )
Assertion lsmelvalmi 
|- ( ph -> ( X .- Y ) e. ( T .(+) U ) )

Proof

Step Hyp Ref Expression
1 lsmelvalm.m 
 |-  .- = ( -g ` G )
2 lsmelvalm.p 
 |-  .(+) = ( LSSum ` G )
3 lsmelvalm.t 
 |-  ( ph -> T e. ( SubGrp ` G ) )
4 lsmelvalm.u 
 |-  ( ph -> U e. ( SubGrp ` G ) )
5 lsmelvalmi.x 
 |-  ( ph -> X e. T )
6 lsmelvalmi.y 
 |-  ( ph -> Y e. U )
7 eqidd 
 |-  ( ph -> ( X .- Y ) = ( X .- Y ) )
8 rspceov 
 |-  ( ( X e. T /\ Y e. U /\ ( X .- Y ) = ( X .- Y ) ) -> E. x e. T E. y e. U ( X .- Y ) = ( x .- y ) )
9 5 6 7 8 syl3anc 
 |-  ( ph -> E. x e. T E. y e. U ( X .- Y ) = ( x .- y ) )
10 1 2 3 4 lsmelvalm 
 |-  ( ph -> ( ( X .- Y ) e. ( T .(+) U ) <-> E. x e. T E. y e. U ( X .- Y ) = ( x .- y ) ) )
11 9 10 mpbird 
 |-  ( ph -> ( X .- Y ) e. ( T .(+) U ) )