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Theorem lshpset 29168
Description: The set of all hyperplanes of a left module or left vector space. The vector  v is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.)
Hypotheses
Ref Expression
lshpset.v  |-  V  =  ( Base `  W
)
lshpset.n  |-  N  =  ( LSpan `  W )
lshpset.s  |-  S  =  ( LSubSp `  W )
lshpset.h  |-  H  =  (LSHyp `  W )
Assertion
Ref Expression
lshpset  |-  ( W  e.  X  ->  H  =  { s  e.  S  |  ( s  =/= 
V  /\  E. v  e.  V  ( N `  ( s  u.  {
v } ) )  =  V ) } )
Distinct variable groups:    S, s    v, V    v, s, W
Allowed substitution hints:    S( v)    H( v, s)    N( v, s)    V( s)    X( v, s)

Proof of Theorem lshpset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 lshpset.h . 2  |-  H  =  (LSHyp `  W )
2 elex 2796 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5525 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
4 lshpset.s . . . . . 6  |-  S  =  ( LSubSp `  W )
53, 4syl6eqr 2333 . . . . 5  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
6 fveq2 5525 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
7 lshpset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
86, 7syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
98neeq2d 2460 . . . . . 6  |-  ( w  =  W  ->  (
s  =/=  ( Base `  w )  <->  s  =/=  V ) )
10 fveq2 5525 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
11 lshpset.n . . . . . . . . . 10  |-  N  =  ( LSpan `  W )
1210, 11syl6eqr 2333 . . . . . . . . 9  |-  ( w  =  W  ->  ( LSpan `  w )  =  N )
1312fveq1d 5527 . . . . . . . 8  |-  ( w  =  W  ->  (
( LSpan `  w ) `  ( s  u.  {
v } ) )  =  ( N `  ( s  u.  {
v } ) ) )
1413, 8eqeq12d 2297 . . . . . . 7  |-  ( w  =  W  ->  (
( ( LSpan `  w
) `  ( s  u.  { v } ) )  =  ( Base `  w )  <->  ( N `  ( s  u.  {
v } ) )  =  V ) )
158, 14rexeqbidv 2749 . . . . . 6  |-  ( w  =  W  ->  ( E. v  e.  ( Base `  w ) ( ( LSpan `  w ) `  ( s  u.  {
v } ) )  =  ( Base `  w
)  <->  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V ) )
169, 15anbi12d 691 . . . . 5  |-  ( w  =  W  ->  (
( s  =/=  ( Base `  w )  /\  E. v  e.  ( Base `  w ) ( (
LSpan `  w ) `  ( s  u.  {
v } ) )  =  ( Base `  w
) )  <->  ( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  {
v } ) )  =  V ) ) )
175, 16rabeqbidv 2783 . . . 4  |-  ( w  =  W  ->  { s  e.  ( LSubSp `  w
)  |  ( s  =/=  ( Base `  w
)  /\  E. v  e.  ( Base `  w
) ( ( LSpan `  w ) `  (
s  u.  { v } ) )  =  ( Base `  w
) ) }  =  { s  e.  S  |  ( s  =/= 
V  /\  E. v  e.  V  ( N `  ( s  u.  {
v } ) )  =  V ) } )
18 df-lshyp 29167 . . . 4  |- LSHyp  =  ( w  e.  _V  |->  { s  e.  ( LSubSp `  w )  |  ( s  =/=  ( Base `  w )  /\  E. v  e.  ( Base `  w ) ( (
LSpan `  w ) `  ( s  u.  {
v } ) )  =  ( Base `  w
) ) } )
19 fvex 5539 . . . . . 6  |-  ( LSubSp `  W )  e.  _V
204, 19eqeltri 2353 . . . . 5  |-  S  e. 
_V
2120rabex 4165 . . . 4  |-  { s  e.  S  |  ( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V ) }  e.  _V
2217, 18, 21fvmpt 5602 . . 3  |-  ( W  e.  _V  ->  (LSHyp `  W )  =  {
s  e.  S  | 
( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V ) } )
232, 22syl 15 . 2  |-  ( W  e.  X  ->  (LSHyp `  W )  =  {
s  e.  S  | 
( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V ) } )
241, 23syl5eq 2327 1  |-  ( W  e.  X  ->  H  =  { s  e.  S  |  ( s  =/= 
V  /\  E. v  e.  V  ( N `  ( s  u.  {
v } ) )  =  V ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   _Vcvv 2788    u. cun 3150   {csn 3640   ` cfv 5255   Basecbs 13148   LSubSpclss 15689   LSpanclspn 15728  LSHypclsh 29165
This theorem is referenced by:  islshp  29169
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-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-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-lshyp 29167
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