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Theorem lsscl 15716
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 2296 . . . 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 15708 . . 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 5881 . . . . 5  |-  ( x  =  Z  ->  (
x  .x.  a )  =  ( Z  .x.  a ) )
109oveq1d 5889 . . . 4  |-  ( x  =  Z  ->  (
( x  .x.  a
)  .+  b )  =  ( ( Z 
.x.  a )  .+  b ) )
1110eleq1d 2362 . . 3  |-  ( x  =  Z  ->  (
( ( x  .x.  a )  .+  b
)  e.  U  <->  ( ( Z  .x.  a )  .+  b )  e.  U
) )
12 oveq2 5882 . . . . 5  |-  ( a  =  X  ->  ( Z  .x.  a )  =  ( Z  .x.  X
) )
1312oveq1d 5889 . . . 4  |-  ( a  =  X  ->  (
( Z  .x.  a
)  .+  b )  =  ( ( Z 
.x.  X )  .+  b ) )
1413eleq1d 2362 . . 3  |-  ( a  =  X  ->  (
( ( Z  .x.  a )  .+  b
)  e.  U  <->  ( ( Z  .x.  X )  .+  b )  e.  U
) )
15 oveq2 5882 . . . 4  |-  ( b  =  Y  ->  (
( Z  .x.  X
)  .+  b )  =  ( ( Z 
.x.  X )  .+  Y ) )
1615eleq1d 2362 . . 3  |-  ( b  =  Y  ->  (
( ( Z  .x.  X )  .+  b
)  e.  U  <->  ( ( Z  .x.  X )  .+  Y )  e.  U
) )
1711, 14, 16rspc3v 2906 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   LSubSpclss 15705
This theorem is referenced by:  lssvsubcl  15717  lssvacl  15727  lssvscl  15728  islss3  15732  lssintcl  15737  lspsolvlem  15911  lbsextlem2  15928  isphld  16574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-lss 15706
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