Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islshp Structured version   Visualization version   Unicode version

Theorem islshp 32545
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 32544 . . 3  |-  ( W  e.  X  ->  H  =  { s  e.  S  |  ( s  =/= 
V  /\  E. v  e.  V  ( N `  ( s  u.  {
v } ) )  =  V ) } )
65eleq2d 2514 . 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 2686 . . . . 5  |-  ( s  =  U  ->  (
s  =/=  V  <->  U  =/=  V ) )
8 uneq1 3581 . . . . . . . 8  |-  ( s  =  U  ->  (
s  u.  { v } )  =  ( U  u.  { v } ) )
98fveq2d 5869 . . . . . . 7  |-  ( s  =  U  ->  ( N `  ( s  u.  { v } ) )  =  ( N `
 ( U  u.  { v } ) ) )
109eqeq1d 2453 . . . . . 6  |-  ( s  =  U  ->  (
( N `  (
s  u.  { v } ) )  =  V  <->  ( N `  ( U  u.  { v } ) )  =  V ) )
1110rexbidv 2901 . . . . 5  |-  ( s  =  U  ->  ( E. v  e.  V  ( N `  ( s  u.  { v } ) )  =  V  <->  E. v  e.  V  ( N `  ( U  u.  { v } ) )  =  V ) )
127, 11anbi12d 717 . . . 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 3196 . . 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 989 . . 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 256 . 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 265 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   {crab 2741    u. cun 3402   {csn 3968   ` cfv 5582   Basecbs 15121   LSubSpclss 18155   LSpanclspn 18194  LSHypclsh 32541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-lshyp 32543
This theorem is referenced by:  islshpsm  32546  lshplss  32547  lshpne  32548  lshpnel2N  32551  lkrshp  32671  lshpset2N  32685  dochsatshp  35019
  Copyright terms: Public domain W3C validator