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Theorem islshp 32464
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 32463 . . 3  |-  ( W  e.  X  ->  H  =  { s  e.  S  |  ( s  =/= 
V  /\  E. v  e.  V  ( N `  ( s  u.  {
v } ) )  =  V ) } )
65eleq2d 2505 . 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 2611 . . . . 5  |-  ( s  =  U  ->  (
s  =/=  V  <->  U  =/=  V ) )
8 uneq1 3498 . . . . . . . 8  |-  ( s  =  U  ->  (
s  u.  { v } )  =  ( U  u.  { v } ) )
98fveq2d 5690 . . . . . . 7  |-  ( s  =  U  ->  ( N `  ( s  u.  { v } ) )  =  ( N `
 ( U  u.  { v } ) ) )
109eqeq1d 2446 . . . . . 6  |-  ( s  =  U  ->  (
( N `  (
s  u.  { v } ) )  =  V  <->  ( N `  ( U  u.  { v } ) )  =  V ) )
1110rexbidv 2731 . . . . 5  |-  ( s  =  U  ->  ( E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V  <->  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V ) )
127, 11anbi12d 710 . . . 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 3112 . . 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 969 . . 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 252 . 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 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711   {crab 2714    u. cun 3321   {csn 3872   ` cfv 5413   Basecbs 14166   LSubSpclss 16990   LSpanclspn 17029  LSHypclsh 32460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-lshyp 32462
This theorem is referenced by:  islshpsm  32465  lshplss  32466  lshpne  32467  lshpnel2N  32470  lkrshp  32590  lshpset2N  32604  dochsatshp  34936
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