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Theorem lsscl 16021
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 2438 . . . 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 16013 . . 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 975 . 2  |-  ( U  e.  S  ->  A. x  e.  B  A. a  e.  U  A. b  e.  U  ( (
x  .x.  a )  .+  b )  e.  U
)
9 oveq1 6090 . . . . 5  |-  ( x  =  Z  ->  (
x  .x.  a )  =  ( Z  .x.  a ) )
109oveq1d 6098 . . . 4  |-  ( x  =  Z  ->  (
( x  .x.  a
)  .+  b )  =  ( ( Z 
.x.  a )  .+  b ) )
1110eleq1d 2504 . . 3  |-  ( x  =  Z  ->  (
( ( x  .x.  a )  .+  b
)  e.  U  <->  ( ( Z  .x.  a )  .+  b )  e.  U
) )
12 oveq2 6091 . . . . 5  |-  ( a  =  X  ->  ( Z  .x.  a )  =  ( Z  .x.  X
) )
1312oveq1d 6098 . . . 4  |-  ( a  =  X  ->  (
( Z  .x.  a
)  .+  b )  =  ( ( Z 
.x.  X )  .+  b ) )
1413eleq1d 2504 . . 3  |-  ( a  =  X  ->  (
( ( Z  .x.  a )  .+  b
)  e.  U  <->  ( ( Z  .x.  X )  .+  b )  e.  U
) )
15 oveq2 6091 . . . 4  |-  ( b  =  Y  ->  (
( Z  .x.  X
)  .+  b )  =  ( ( Z 
.x.  X )  .+  Y ) )
1615eleq1d 2504 . . 3  |-  ( b  =  Y  ->  (
( ( Z  .x.  X )  .+  b
)  e.  U  <->  ( ( Z  .x.  X )  .+  Y )  e.  U
) )
1711, 14, 16rspc3v 3063 . 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 457 1  |-  ( ( U  e.  S  /\  ( Z  e.  B  /\  X  e.  U  /\  Y  e.  U
) )  ->  (
( Z  .x.  X
)  .+  Y )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707    C_ wss 3322   (/)c0 3630   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531  Scalarcsca 13534   .scvsca 13535   LSubSpclss 16010
This theorem is referenced by:  lssvsubcl  16022  lssvacl  16032  lssvscl  16033  islss3  16037  lssintcl  16042  lspsolvlem  16216  lbsextlem2  16233  isphld  16887
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-lss 16011
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