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Metamath Proof Explorer

Theorem lsmelvalx

Description: Subspace sum membership (for a group or vector space). Extended domain version of lsmelval . (Contributed by NM, 28-Jan-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 lsmelvalx 
|- ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( X e. ( T .(+) U ) <-> E. y e. T E. z e. U X = ( y .+ z ) ) )

Proof

Step Hyp Ref Expression
1 lsmfval.v 
 |-  B = ( Base ` G )
2 lsmfval.a 
 |-  .+ = ( +g ` G )
3 lsmfval.s 
 |-  .(+) = ( LSSum ` G )
4 1 2 3 lsmvalx 
 |-  ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( T .(+) U ) = ran ( y e. T , z e. U |-> ( y .+ z ) ) )
5 4 eleq2d 
 |-  ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( X e. ( T .(+) U ) <-> X e. ran ( y e. T , z e. U |-> ( y .+ z ) ) ) )
6 eqid 
 |-  ( y e. T , z e. U |-> ( y .+ z ) ) = ( y e. T , z e. U |-> ( y .+ z ) )
7 ovex 
 |-  ( y .+ z ) e. _V
8 6 7 elrnmpo 
 |-  ( X e. ran ( y e. T , z e. U |-> ( y .+ z ) ) <-> E. y e. T E. z e. U X = ( y .+ z ) )
9 5 8 bitrdi 
 |-  ( ( G e. V /\ T C_ B /\ U C_ B ) -> ( X e. ( T .(+) U ) <-> E. y e. T E. z e. U X = ( y .+ z ) ) )