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Theorem lssset 15691
Description: The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)
Hypotheses
Ref Expression
lssset.f  |-  F  =  (Scalar `  W )
lssset.b  |-  B  =  ( Base `  F
)
lssset.v  |-  V  =  ( Base `  W
)
lssset.p  |-  .+  =  ( +g  `  W )
lssset.t  |-  .x.  =  ( .s `  W )
lssset.s  |-  S  =  ( LSubSp `  W )
Assertion
Ref Expression
lssset  |-  ( W  e.  X  ->  S  =  { s  e.  ( ~P V  \  { (/)
} )  |  A. x  e.  B  A. a  e.  s  A. b  e.  s  (
( x  .x.  a
)  .+  b )  e.  s } )
Distinct variable groups:    .+ , s    x, s, B    V, s    a,
b, s, x, W    .x. , s
Allowed substitution hints:    B( a, b)    .+ ( x, a, b)    S( x, s, a, b)    .x. ( x, a, b)    F( x, s, a, b)    V( x, a, b)    X( x, s, a, b)

Proof of Theorem lssset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 lssset.s . 2  |-  S  =  ( LSubSp `  W )
2 elex 2796 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5525 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lssset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
65pweqd 3630 . . . . . 6  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
76difeq1d 3293 . . . . 5  |-  ( w  =  W  ->  ( ~P ( Base `  w
)  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
8 fveq2 5525 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
9 lssset.f . . . . . . . . 9  |-  F  =  (Scalar `  W )
108, 9syl6eqr 2333 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
1110fveq2d 5529 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
12 lssset.b . . . . . . 7  |-  B  =  ( Base `  F
)
1311, 12syl6eqr 2333 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  B )
14 fveq2 5525 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
15 lssset.t . . . . . . . . . . . 12  |-  .x.  =  ( .s `  W )
1614, 15syl6eqr 2333 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
1716oveqd 5875 . . . . . . . . . 10  |-  ( w  =  W  ->  (
x ( .s `  w ) a )  =  ( x  .x.  a ) )
1817oveq1d 5873 . . . . . . . . 9  |-  ( w  =  W  ->  (
( x ( .s
`  w ) a ) ( +g  `  w
) b )  =  ( ( x  .x.  a ) ( +g  `  w ) b ) )
19 fveq2 5525 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  w )  =  ( +g  `  W
) )
20 lssset.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  W )
2119, 20syl6eqr 2333 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  w )  = 
.+  )
2221oveqd 5875 . . . . . . . . 9  |-  ( w  =  W  ->  (
( x  .x.  a
) ( +g  `  w
) b )  =  ( ( x  .x.  a )  .+  b
) )
2318, 22eqtrd 2315 . . . . . . . 8  |-  ( w  =  W  ->  (
( x ( .s
`  w ) a ) ( +g  `  w
) b )  =  ( ( x  .x.  a )  .+  b
) )
2423eleq1d 2349 . . . . . . 7  |-  ( w  =  W  ->  (
( ( x ( .s `  w ) a ) ( +g  `  w ) b )  e.  s  <->  ( (
x  .x.  a )  .+  b )  e.  s ) )
25242ralbidv 2585 . . . . . 6  |-  ( w  =  W  ->  ( A. a  e.  s  A. b  e.  s 
( ( x ( .s `  w ) a ) ( +g  `  w ) b )  e.  s  <->  A. a  e.  s  A. b  e.  s  ( (
x  .x.  a )  .+  b )  e.  s ) )
2613, 25raleqbidv 2748 . . . . 5  |-  ( w  =  W  ->  ( A. x  e.  ( Base `  (Scalar `  w
) ) A. a  e.  s  A. b  e.  s  ( (
x ( .s `  w ) a ) ( +g  `  w
) b )  e.  s  <->  A. x  e.  B  A. a  e.  s  A. b  e.  s 
( ( x  .x.  a )  .+  b
)  e.  s ) )
277, 26rabeqbidv 2783 . . . 4  |-  ( w  =  W  ->  { s  e.  ( ~P ( Base `  w )  \  { (/) } )  | 
A. x  e.  (
Base `  (Scalar `  w
) ) A. a  e.  s  A. b  e.  s  ( (
x ( .s `  w ) a ) ( +g  `  w
) b )  e.  s }  =  {
s  e.  ( ~P V  \  { (/) } )  |  A. x  e.  B  A. a  e.  s  A. b  e.  s  ( (
x  .x.  a )  .+  b )  e.  s } )
28 df-lss 15690 . . . 4  |-  LSubSp  =  ( w  e.  _V  |->  { s  e.  ( ~P ( Base `  w
)  \  { (/) } )  |  A. x  e.  ( Base `  (Scalar `  w ) ) A. a  e.  s  A. b  e.  s  (
( x ( .s
`  w ) a ) ( +g  `  w
) b )  e.  s } )
29 fvex 5539 . . . . . . . 8  |-  ( Base `  W )  e.  _V
304, 29eqeltri 2353 . . . . . . 7  |-  V  e. 
_V
3130pwex 4193 . . . . . 6  |-  ~P V  e.  _V
32 difexg 4162 . . . . . 6  |-  ( ~P V  e.  _V  ->  ( ~P V  \  { (/)
} )  e.  _V )
3331, 32ax-mp 8 . . . . 5  |-  ( ~P V  \  { (/) } )  e.  _V
3433rabex 4165 . . . 4  |-  { s  e.  ( ~P V  \  { (/) } )  | 
A. x  e.  B  A. a  e.  s  A. b  e.  s 
( ( x  .x.  a )  .+  b
)  e.  s }  e.  _V
3527, 28, 34fvmpt 5602 . . 3  |-  ( W  e.  _V  ->  ( LSubSp `
 W )  =  { s  e.  ( ~P V  \  { (/)
} )  |  A. x  e.  B  A. a  e.  s  A. b  e.  s  (
( x  .x.  a
)  .+  b )  e.  s } )
362, 35syl 15 . 2  |-  ( W  e.  X  ->  ( LSubSp `
 W )  =  { s  e.  ( ~P V  \  { (/)
} )  |  A. x  e.  B  A. a  e.  s  A. b  e.  s  (
( x  .x.  a
)  .+  b )  e.  s } )
371, 36syl5eq 2327 1  |-  ( W  e.  X  ->  S  =  { s  e.  ( ~P V  \  { (/)
} )  |  A. x  e.  B  A. a  e.  s  A. b  e.  s  (
( x  .x.  a
)  .+  b )  e.  s } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149   (/)c0 3455   ~Pcpw 3625   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   LSubSpclss 15689
This theorem is referenced by:  islss  15692
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-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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