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Theorem lssset 17014
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 2980 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5690 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lssset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2492 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
65pweqd 3864 . . . . . 6  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
76difeq1d 3472 . . . . 5  |-  ( w  =  W  ->  ( ~P ( Base `  w
)  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
8 fveq2 5690 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
9 lssset.f . . . . . . . . 9  |-  F  =  (Scalar `  W )
108, 9syl6eqr 2492 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
1110fveq2d 5694 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
12 lssset.b . . . . . . 7  |-  B  =  ( Base `  F
)
1311, 12syl6eqr 2492 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  B )
14 fveq2 5690 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
15 lssset.t . . . . . . . . . . . 12  |-  .x.  =  ( .s `  W )
1614, 15syl6eqr 2492 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
1716oveqd 6107 . . . . . . . . . 10  |-  ( w  =  W  ->  (
x ( .s `  w ) a )  =  ( x  .x.  a ) )
1817oveq1d 6105 . . . . . . . . 9  |-  ( w  =  W  ->  (
( x ( .s
`  w ) a ) ( +g  `  w
) b )  =  ( ( x  .x.  a ) ( +g  `  w ) b ) )
19 fveq2 5690 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  w )  =  ( +g  `  W
) )
20 lssset.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  W )
2119, 20syl6eqr 2492 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  w )  = 
.+  )
2221oveqd 6107 . . . . . . . . 9  |-  ( w  =  W  ->  (
( x  .x.  a
) ( +g  `  w
) b )  =  ( ( x  .x.  a )  .+  b
) )
2318, 22eqtrd 2474 . . . . . . . 8  |-  ( w  =  W  ->  (
( x ( .s
`  w ) a ) ( +g  `  w
) b )  =  ( ( x  .x.  a )  .+  b
) )
2423eleq1d 2508 . . . . . . 7  |-  ( w  =  W  ->  (
( ( x ( .s `  w ) a ) ( +g  `  w ) b )  e.  s  <->  ( (
x  .x.  a )  .+  b )  e.  s ) )
25242ralbidv 2756 . . . . . 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 2930 . . . . 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 2966 . . . 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 17013 . . . 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 5700 . . . . . . . 8  |-  ( Base `  W )  e.  _V
304, 29eqeltri 2512 . . . . . . 7  |-  V  e. 
_V
3130pwex 4474 . . . . . 6  |-  ~P V  e.  _V
32 difexg 4439 . . . . . 6  |-  ( ~P V  e.  _V  ->  ( ~P V  \  { (/)
} )  e.  _V )
3331, 32ax-mp 5 . . . . 5  |-  ( ~P V  \  { (/) } )  e.  _V
3433rabex 4442 . . . 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 5773 . . 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 16 . 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 2486 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2714   {crab 2718   _Vcvv 2971    \ cdif 3324   (/)c0 3636   ~Pcpw 3859   {csn 3876   ` cfv 5417  (class class class)co 6090   Basecbs 14173   +g cplusg 14237  Scalarcsca 14240   .scvsca 14241   LSubSpclss 17012
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093  df-lss 17013
This theorem is referenced by:  islss  17015
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