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Theorem lshpne 29717
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 2435 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 eqid 2435 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
6 lshpne.h . . . . 5  |-  H  =  (LSHyp `  W )
73, 4, 5, 6islshp 29714 . . . 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 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    u. cun 3310   {csn 3806   ` cfv 5446   Basecbs 13461   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039  LSHypclsh 29710
This theorem is referenced by:  lshpnel  29718  lshpcmp  29723  lkrshp3  29841  lkrshp4  29843  dochshpncl  32119  dochlkr  32120  dochkrshp  32121  dochsatshpb  32187
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-lshyp 29712
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