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Theorem lsscl 17460
Description: Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
Hypotheses
Ref Expression
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
Ref Expression
lsscl  |-  ( ( U  e.  S  /\  ( Z  e.  B  /\  X  e.  U  /\  Y  e.  U
) )  ->  (
( Z  .x.  X
)  .+  Y )  e.  U )

Proof of Theorem lsscl
Dummy variables  x  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsscl.f . . . 4  |-  F  =  (Scalar `  W )
2 lsscl.b . . . 4  |-  B  =  ( Base `  F
)
3 eqid 2467 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
4 lsscl.p . . . 4  |-  .+  =  ( +g  `  W )
5 lsscl.t . . . 4  |-  .x.  =  ( .s `  W )
6 lsscl.s . . . 4  |-  S  =  ( LSubSp `  W )
71, 2, 3, 4, 5, 6islss 17452 . . 3  |-  ( 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 ) )
87simp3bi 1013 . 2  |-  ( U  e.  S  ->  A. x  e.  B  A. a  e.  U  A. b  e.  U  ( (
x  .x.  a )  .+  b )  e.  U
)
9 oveq1 6302 . . . . 5  |-  ( x  =  Z  ->  (
x  .x.  a )  =  ( Z  .x.  a ) )
109oveq1d 6310 . . . 4  |-  ( x  =  Z  ->  (
( x  .x.  a
)  .+  b )  =  ( ( Z 
.x.  a )  .+  b ) )
1110eleq1d 2536 . . 3  |-  ( x  =  Z  ->  (
( ( x  .x.  a )  .+  b
)  e.  U  <->  ( ( Z  .x.  a )  .+  b )  e.  U
) )
12 oveq2 6303 . . . . 5  |-  ( a  =  X  ->  ( Z  .x.  a )  =  ( Z  .x.  X
) )
1312oveq1d 6310 . . . 4  |-  ( a  =  X  ->  (
( Z  .x.  a
)  .+  b )  =  ( ( Z 
.x.  X )  .+  b ) )
1413eleq1d 2536 . . 3  |-  ( a  =  X  ->  (
( ( Z  .x.  a )  .+  b
)  e.  U  <->  ( ( Z  .x.  X )  .+  b )  e.  U
) )
15 oveq2 6303 . . . 4  |-  ( b  =  Y  ->  (
( Z  .x.  X
)  .+  b )  =  ( ( Z 
.x.  X )  .+  Y ) )
1615eleq1d 2536 . . 3  |-  ( b  =  Y  ->  (
( ( Z  .x.  X )  .+  b
)  e.  U  <->  ( ( Z  .x.  X )  .+  Y )  e.  U
) )
1711, 14, 16rspc3v 3231 . 2  |-  ( ( 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 ) )
188, 17mpan9 469 1  |-  ( ( U  e.  S  /\  ( Z  e.  B  /\  X  e.  U  /\  Y  e.  U
) )  ->  (
( Z  .x.  X
)  .+  Y )  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817    C_ wss 3481   (/)c0 3790   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572  Scalarcsca 14575   .scvsca 14576   LSubSpclss 17449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-lss 17450
This theorem is referenced by:  lssvsubcl  17461  lssvacl  17471  lssvscl  17472  islss3  17476  lssintcl  17481  lspsolvlem  17659  lbsextlem2  17676  isphld  18558
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