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Theorem islshp 29791
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 29790 . . 3  |-  ( W  e.  X  ->  H  =  { s  e.  S  |  ( s  =/= 
V  /\  E. v  e.  V  ( N `  ( s  u.  {
v } ) )  =  V ) } )
65eleq2d 2363 . 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 2467 . . . . 5  |-  ( s  =  U  ->  (
s  =/=  V  <->  U  =/=  V ) )
8 uneq1 3335 . . . . . . . 8  |-  ( s  =  U  ->  (
s  u.  { v } )  =  ( U  u.  { v } ) )
98fveq2d 5545 . . . . . . 7  |-  ( s  =  U  ->  ( N `  ( s  u.  { v } ) )  =  ( N `
 ( U  u.  { v } ) ) )
109eqeq1d 2304 . . . . . 6  |-  ( s  =  U  ->  (
( N `  (
s  u.  { v } ) )  =  V  <->  ( N `  ( U  u.  { v } ) )  =  V ) )
1110rexbidv 2577 . . . . 5  |-  ( s  =  U  ->  ( E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V  <->  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V ) )
127, 11anbi12d 691 . . . 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 2936 . . 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 938 . . 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 243 . 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 252 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {crab 2560    u. cun 3163   {csn 3653   ` cfv 5271   Basecbs 13164   LSubSpclss 15705   LSpanclspn 15744  LSHypclsh 29787
This theorem is referenced by:  islshpsm  29792  lshplss  29793  lshpne  29794  lshpnel2N  29797  lkrshp  29917  lshpset2N  29931  dochsatshp  32263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-lshyp 29789
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