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Theorem lsscl 17024
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 2443 . . . 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 17016 . . 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 1005 . 2  |-  ( U  e.  S  ->  A. x  e.  B  A. a  e.  U  A. b  e.  U  ( (
x  .x.  a )  .+  b )  e.  U
)
9 oveq1 6098 . . . . 5  |-  ( x  =  Z  ->  (
x  .x.  a )  =  ( Z  .x.  a ) )
109oveq1d 6106 . . . 4  |-  ( x  =  Z  ->  (
( x  .x.  a
)  .+  b )  =  ( ( Z 
.x.  a )  .+  b ) )
1110eleq1d 2509 . . 3  |-  ( x  =  Z  ->  (
( ( x  .x.  a )  .+  b
)  e.  U  <->  ( ( Z  .x.  a )  .+  b )  e.  U
) )
12 oveq2 6099 . . . . 5  |-  ( a  =  X  ->  ( Z  .x.  a )  =  ( Z  .x.  X
) )
1312oveq1d 6106 . . . 4  |-  ( a  =  X  ->  (
( Z  .x.  a
)  .+  b )  =  ( ( Z 
.x.  X )  .+  b ) )
1413eleq1d 2509 . . 3  |-  ( a  =  X  ->  (
( ( Z  .x.  a )  .+  b
)  e.  U  <->  ( ( Z  .x.  X )  .+  b )  e.  U
) )
15 oveq2 6099 . . . 4  |-  ( b  =  Y  ->  (
( Z  .x.  X
)  .+  b )  =  ( ( Z 
.x.  X )  .+  Y ) )
1615eleq1d 2509 . . 3  |-  ( b  =  Y  ->  (
( ( Z  .x.  X )  .+  b
)  e.  U  <->  ( ( Z  .x.  X )  .+  Y )  e.  U
) )
1711, 14, 16rspc3v 3082 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715    C_ wss 3328   (/)c0 3637   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238  Scalarcsca 14241   .scvsca 14242   LSubSpclss 17013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-lss 17014
This theorem is referenced by:  lssvsubcl  17025  lssvacl  17035  lssvscl  17036  islss3  17040  lssintcl  17045  lspsolvlem  17223  lbsextlem2  17240  isphld  18083
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