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Theorem islshp 29094
Description: The predicate "is a hyperplane" (of a left module or left vector space). (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
islshp  |-  ( W  e.  X  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/= 
V  /\  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V ) ) )
Distinct variable groups:    v, V    v, W    v, U
Allowed substitution hints:    S( v)    H( v)    N( v)    X( v)

Proof of Theorem islshp
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 lshpset.v . . . 4  |-  V  =  ( Base `  W
)
2 lshpset.n . . . 4  |-  N  =  ( LSpan `  W )
3 lshpset.s . . . 4  |-  S  =  ( LSubSp `  W )
4 lshpset.h . . . 4  |-  H  =  (LSHyp `  W )
51, 2, 3, 4lshpset 29093 . . 3  |-  ( W  e.  X  ->  H  =  { s  e.  S  |  ( s  =/= 
V  /\  E. v  e.  V  ( N `  ( s  u.  {
v } ) )  =  V ) } )
65eleq2d 2454 . 2  |-  ( W  e.  X  ->  ( U  e.  H  <->  U  e.  { s  e.  S  | 
( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V ) } ) )
7 neeq1 2558 . . . . 5  |-  ( s  =  U  ->  (
s  =/=  V  <->  U  =/=  V ) )
8 uneq1 3437 . . . . . . . 8  |-  ( s  =  U  ->  (
s  u.  { v } )  =  ( U  u.  { v } ) )
98fveq2d 5672 . . . . . . 7  |-  ( s  =  U  ->  ( N `  ( s  u.  { v } ) )  =  ( N `
 ( U  u.  { v } ) ) )
109eqeq1d 2395 . . . . . 6  |-  ( s  =  U  ->  (
( N `  (
s  u.  { v } ) )  =  V  <->  ( N `  ( U  u.  { v } ) )  =  V ) )
1110rexbidv 2670 . . . . 5  |-  ( s  =  U  ->  ( E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V  <->  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V ) )
127, 11anbi12d 692 . . . 4  |-  ( s  =  U  ->  (
( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V )  <->  ( U  =/= 
V  /\  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V ) ) )
1312elrab 3035 . . 3  |-  ( U  e.  { s  e.  S  |  ( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V ) }  <->  ( U  e.  S  /\  ( U  =/=  V  /\  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V ) ) )
14 3anass 940 . . 3  |-  ( ( U  e.  S  /\  U  =/=  V  /\  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V )  <-> 
( U  e.  S  /\  ( U  =/=  V  /\  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V ) ) )
1513, 14bitr4i 244 . 2  |-  ( U  e.  { s  e.  S  |  ( s  =/=  V  /\  E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V ) }  <->  ( U  e.  S  /\  U  =/= 
V  /\  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V ) )
166, 15syl6bb 253 1  |-  ( W  e.  X  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/= 
V  /\  E. v  e.  V  ( N `  ( U  u.  {
v } ) )  =  V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650   {crab 2653    u. cun 3261   {csn 3757   ` cfv 5394   Basecbs 13396   LSubSpclss 15935   LSpanclspn 15974  LSHypclsh 29090
This theorem is referenced by:  islshpsm  29095  lshplss  29096  lshpne  29097  lshpnel2N  29100  lkrshp  29220  lshpset2N  29234  dochsatshp  31566
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-lshyp 29092
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