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

Metamath Proof Explorer

Theorem lsmelvalix

Description: Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-2014) (Revised by Mario Carneiro, 19-Apr-2016)

Ref Expression
Hypotheses lsmfval.v 
|- B = ( Base ` G )
lsmfval.a 
|- .+ = ( +g ` G )
lsmfval.s 
|- .(+) = ( LSSum ` G )
Assertion lsmelvalix 
|- ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( X e. T /\ Y e. U ) ) -> ( X .+ Y ) e. ( T .(+) U ) )

Proof

Step Hyp Ref Expression
1 lsmfval.v 
 |-  B = ( Base ` G )
2 lsmfval.a 
 |-  .+ = ( +g ` G )
3 lsmfval.s 
 |-  .(+) = ( LSSum ` G )
4 eqid 
 |-  ( X .+ Y ) = ( X .+ Y )
5 rspceov 
 |-  ( ( X e. T /\ Y e. U /\ ( X .+ Y ) = ( X .+ Y ) ) -> E. x e. T E. y e. U ( X .+ Y ) = ( x .+ y ) )
6 4 5 mp3an3 
 |-  ( ( X e. T /\ Y e. U ) -> E. x e. T E. y e. U ( X .+ Y ) = ( x .+ y ) )
7 1 2 3 lsmelvalx 
 |-  ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( ( X .+ Y ) e. ( T .(+) U ) <-> E. x e. T E. y e. U ( X .+ Y ) = ( x .+ y ) ) )
8 7 biimpar 
 |-  ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ E. x e. T E. y e. U ( X .+ Y ) = ( x .+ y ) ) -> ( X .+ Y ) e. ( T .(+) U ) )
9 6 8 sylan2 
 |-  ( ( ( G e. V /\ T C_ B /\ U C_ B ) /\ ( X e. T /\ Y e. U ) ) -> ( X .+ Y ) e. ( T .(+) U ) )