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Theorem lshpne 29097
Description: A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
Hypotheses
Ref Expression
lshpne.v  |-  V  =  ( Base `  W
)
lshpne.h  |-  H  =  (LSHyp `  W )
lshpne.w  |-  ( ph  ->  W  e.  LMod )
lshpne.u  |-  ( ph  ->  U  e.  H )
Assertion
Ref Expression
lshpne  |-  ( ph  ->  U  =/=  V )

Proof of Theorem lshpne
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lshpne.u . . 3  |-  ( ph  ->  U  e.  H )
2 lshpne.w . . . 4  |-  ( ph  ->  W  e.  LMod )
3 lshpne.v . . . . 5  |-  V  =  ( Base `  W
)
4 eqid 2387 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 eqid 2387 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
6 lshpne.h . . . . 5  |-  H  =  (LSHyp `  W )
73, 4, 5, 6islshp 29094 . . . 4  |-  ( W  e.  LMod  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W
) `  ( U  u.  { v } ) )  =  V ) ) )
82, 7syl 16 . . 3  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( U  u.  { v } ) )  =  V ) ) )
91, 8mpbid 202 . 2  |-  ( ph  ->  ( U  e.  (
LSubSp `  W )  /\  U  =/=  V  /\  E. v  e.  V  (
( LSpan `  W ) `  ( U  u.  {
v } ) )  =  V ) )
109simp2d 970 1  |-  ( ph  ->  U  =/=  V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650    u. cun 3261   {csn 3757   ` cfv 5394   Basecbs 13396   LModclmod 15877   LSubSpclss 15935   LSpanclspn 15974  LSHypclsh 29090
This theorem is referenced by:  lshpnel  29098  lshpcmp  29103  lkrshp3  29221  lkrshp4  29223  dochshpncl  31499  dochlkr  31500  dochkrshp  31501  dochsatshpb  31567
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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