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Theorem lssset 17706
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 3118 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
4 lssset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
53, 4syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
65pweqd 4020 . . . . . 6  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
76difeq1d 3617 . . . . 5  |-  ( w  =  W  ->  ( ~P ( Base `  w
)  \  { (/) } )  =  ( ~P V  \  { (/) } ) )
8 fveq2 5872 . . . . . . . . 9  |-  ( w  =  W  ->  (Scalar `  w )  =  (Scalar `  W ) )
9 lssset.f . . . . . . . . 9  |-  F  =  (Scalar `  W )
108, 9syl6eqr 2516 . . . . . . . 8  |-  ( w  =  W  ->  (Scalar `  w )  =  F )
1110fveq2d 5876 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  (
Base `  F )
)
12 lssset.b . . . . . . 7  |-  B  =  ( Base `  F
)
1311, 12syl6eqr 2516 . . . . . 6  |-  ( w  =  W  ->  ( Base `  (Scalar `  w
) )  =  B )
14 fveq2 5872 . . . . . . . . . . . 12  |-  ( w  =  W  ->  ( .s `  w )  =  ( .s `  W
) )
15 lssset.t . . . . . . . . . . . 12  |-  .x.  =  ( .s `  W )
1614, 15syl6eqr 2516 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( .s `  w )  = 
.x.  )
1716oveqd 6313 . . . . . . . . . 10  |-  ( w  =  W  ->  (
x ( .s `  w ) a )  =  ( x  .x.  a ) )
1817oveq1d 6311 . . . . . . . . 9  |-  ( w  =  W  ->  (
( x ( .s
`  w ) a ) ( +g  `  w
) b )  =  ( ( x  .x.  a ) ( +g  `  w ) b ) )
19 fveq2 5872 . . . . . . . . . . 11  |-  ( w  =  W  ->  ( +g  `  w )  =  ( +g  `  W
) )
20 lssset.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  W )
2119, 20syl6eqr 2516 . . . . . . . . . 10  |-  ( w  =  W  ->  ( +g  `  w )  = 
.+  )
2221oveqd 6313 . . . . . . . . 9  |-  ( w  =  W  ->  (
( x  .x.  a
) ( +g  `  w
) b )  =  ( ( x  .x.  a )  .+  b
) )
2318, 22eqtrd 2498 . . . . . . . 8  |-  ( w  =  W  ->  (
( x ( .s
`  w ) a ) ( +g  `  w
) b )  =  ( ( x  .x.  a )  .+  b
) )
2423eleq1d 2526 . . . . . . 7  |-  ( w  =  W  ->  (
( ( x ( .s `  w ) a ) ( +g  `  w ) b )  e.  s  <->  ( (
x  .x.  a )  .+  b )  e.  s ) )
25242ralbidv 2901 . . . . . 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 3068 . . . . 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 3104 . . . 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 17705 . . . 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 5882 . . . . . . . 8  |-  ( Base `  W )  e.  _V
304, 29eqeltri 2541 . . . . . . 7  |-  V  e. 
_V
3130pwex 4639 . . . . . 6  |-  ~P V  e.  _V
32 difexg 4604 . . . . . 6  |-  ( ~P V  e.  _V  ->  ( ~P V  \  { (/)
} )  e.  _V )
3331, 32ax-mp 5 . . . . 5  |-  ( ~P V  \  { (/) } )  e.  _V
3433rabex 4607 . . . 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 5956 . . 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 2510 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 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    \ cdif 3468   (/)c0 3793   ~Pcpw 4015   {csn 4032   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711  Scalarcsca 14714   .scvsca 14715   LSubSpclss 17704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-lss 17705
This theorem is referenced by:  islss  17707
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