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Metamath Proof Explorer

Theorem lsscl

Description: Closure property of a subspace. (Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 8-Jan-2015)

Ref Expression
Hypotheses lsscl.f 
|- F = ( Scalar ` W )
lsscl.b 
|- B = ( Base ` F )
lsscl.p 
|- .+ = ( +g ` W )
lsscl.t 
|- .x. = ( .s ` W )
lsscl.s 
|- S = ( LSubSp ` W )
Assertion lsscl 
|- ( ( U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( ( Z .x. X ) .+ Y ) e. U )

Proof

Step Hyp Ref Expression
1 lsscl.f 
 |-  F = ( Scalar ` W )
2 lsscl.b 
 |-  B = ( Base ` F )
3 lsscl.p 
 |-  .+ = ( +g ` W )
4 lsscl.t 
 |-  .x. = ( .s ` W )
5 lsscl.s 
 |-  S = ( LSubSp ` W )
6 eqid 
 |-  ( Base ` W ) = ( Base ` W )
7 1 2 6 3 4 5 islss 
 |-  ( U e. S <-> ( U C_ ( Base ` W ) /\ U =/= (/) /\ A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) )
8 7 simp3bi 
 |-  ( U e. S -> A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U )
9 oveq1 
 |-  ( x = Z -> ( x .x. a ) = ( Z .x. a ) )
10 9 oveq1d 
 |-  ( x = Z -> ( ( x .x. a ) .+ b ) = ( ( Z .x. a ) .+ b ) )
11 10 eleq1d 
 |-  ( x = Z -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( Z .x. a ) .+ b ) e. U ) )
12 oveq2 
 |-  ( a = X -> ( Z .x. a ) = ( Z .x. X ) )
13 12 oveq1d 
 |-  ( a = X -> ( ( Z .x. a ) .+ b ) = ( ( Z .x. X ) .+ b ) )
14 13 eleq1d 
 |-  ( a = X -> ( ( ( Z .x. a ) .+ b ) e. U <-> ( ( Z .x. X ) .+ b ) e. U ) )
15 oveq2 
 |-  ( b = Y -> ( ( Z .x. X ) .+ b ) = ( ( Z .x. X ) .+ Y ) )
16 15 eleq1d 
 |-  ( b = Y -> ( ( ( Z .x. X ) .+ b ) e. U <-> ( ( Z .x. X ) .+ Y ) e. U ) )
17 11 14 16 rspc3v 
 |-  ( ( Z e. B /\ X e. U /\ Y e. U ) -> ( A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U -> ( ( Z .x. X ) .+ Y ) e. U ) )
18 8 17 mpan9 
 |-  ( ( U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( ( Z .x. X ) .+ Y ) e. U )