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Theorem lshpset 29874
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 2970 . . 3  |-  ( W  e.  X  ->  W  e.  _V )
3 fveq2 5757 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
4 lshpset.s . . . . . 6  |-  S  =  ( LSubSp `  W )
53, 4syl6eqr 2492 . . . . 5  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
6 fveq2 5757 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
7 lshpset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
86, 7syl6eqr 2492 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
98neeq2d 2621 . . . . . 6  |-  ( w  =  W  ->  (
s  =/=  ( Base `  w )  <->  s  =/=  V ) )
10 fveq2 5757 . . . . . . . . . 10  |-  ( w  =  W  ->  ( LSpan `  w )  =  ( LSpan `  W )
)
11 lshpset.n . . . . . . . . . 10  |-  N  =  ( LSpan `  W )
1210, 11syl6eqr 2492 . . . . . . . . 9  |-  ( w  =  W  ->  ( LSpan `  w )  =  N )
1312fveq1d 5759 . . . . . . . 8  |-  ( w  =  W  ->  (
( LSpan `  w ) `  ( s  u.  {
v } ) )  =  ( N `  ( s  u.  {
v } ) ) )
1413, 8eqeq12d 2456 . . . . . . 7  |-  ( w  =  W  ->  (
( ( LSpan `  w
) `  ( s  u.  { v } ) )  =  ( Base `  w )  <->  ( N `  ( s  u.  {
v } ) )  =  V ) )
158, 14rexeqbidv 2923 . . . . . 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 693 . . . . 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 2957 . . . 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 29873 . . . 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 5771 . . . . . 6  |-  ( LSubSp `  W )  e.  _V
204, 19eqeltri 2512 . . . . 5  |-  S  e. 
_V
2120rabex 4383 . . . 4  |-  { s  e.  S  |  ( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V ) }  e.  _V
2217, 18, 21fvmpt 5835 . . 3  |-  ( W  e.  _V  ->  (LSHyp `  W )  =  {
s  e.  S  | 
( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V ) } )
232, 22syl 16 . 2  |-  ( W  e.  X  ->  (LSHyp `  W )  =  {
s  e.  S  | 
( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V ) } )
241, 23syl5eq 2486 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 360    = wceq 1653    e. wcel 1727    =/= wne 2605   E.wrex 2712   {crab 2715   _Vcvv 2962    u. cun 3304   {csn 3838   ` cfv 5483   Basecbs 13500   LSubSpclss 16039   LSpanclspn 16078  LSHypclsh 29871
This theorem is referenced by:  islshp  29875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-lshyp 29873
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