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Theorem lsscl 15700
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 2283 . . . 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 15692 . . 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 972 . 2  |-  ( U  e.  S  ->  A. x  e.  B  A. a  e.  U  A. b  e.  U  ( (
x  .x.  a )  .+  b )  e.  U
)
9 oveq1 5865 . . . . 5  |-  ( x  =  Z  ->  (
x  .x.  a )  =  ( Z  .x.  a ) )
109oveq1d 5873 . . . 4  |-  ( x  =  Z  ->  (
( x  .x.  a
)  .+  b )  =  ( ( Z 
.x.  a )  .+  b ) )
1110eleq1d 2349 . . 3  |-  ( x  =  Z  ->  (
( ( x  .x.  a )  .+  b
)  e.  U  <->  ( ( Z  .x.  a )  .+  b )  e.  U
) )
12 oveq2 5866 . . . . 5  |-  ( a  =  X  ->  ( Z  .x.  a )  =  ( Z  .x.  X
) )
1312oveq1d 5873 . . . 4  |-  ( a  =  X  ->  (
( Z  .x.  a
)  .+  b )  =  ( ( Z 
.x.  X )  .+  b ) )
1413eleq1d 2349 . . 3  |-  ( a  =  X  ->  (
( ( Z  .x.  a )  .+  b
)  e.  U  <->  ( ( Z  .x.  X )  .+  b )  e.  U
) )
15 oveq2 5866 . . . 4  |-  ( b  =  Y  ->  (
( Z  .x.  X
)  .+  b )  =  ( ( Z 
.x.  X )  .+  Y ) )
1615eleq1d 2349 . . 3  |-  ( b  =  Y  ->  (
( ( Z  .x.  X )  .+  b
)  e.  U  <->  ( ( Z  .x.  X )  .+  Y )  e.  U
) )
1711, 14, 16rspc3v 2893 . 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 455 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    C_ wss 3152   (/)c0 3455   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   LSubSpclss 15689
This theorem is referenced by:  lssvsubcl  15701  lssvacl  15711  lssvscl  15712  islss3  15716  lssintcl  15721  lspsolvlem  15895  lbsextlem2  15912  isphld  16558
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-lss 15690
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