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Theorem lshplss 35103
Description: A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
Hypotheses
Ref Expression
lshplss.s  |-  S  =  ( LSubSp `  W )
lshplss.h  |-  H  =  (LSHyp `  W )
lshplss.w  |-  ( ph  ->  W  e.  LMod )
lshplss.u  |-  ( ph  ->  U  e.  H )
Assertion
Ref Expression
lshplss  |-  ( ph  ->  U  e.  S )

Proof of Theorem lshplss
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lshplss.u . . 3  |-  ( ph  ->  U  e.  H )
2 lshplss.w . . . 4  |-  ( ph  ->  W  e.  LMod )
3 eqid 2454 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2454 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lshplss.s . . . . 5  |-  S  =  ( LSubSp `  W )
6 lshplss.h . . . . 5  |-  H  =  (LSHyp `  W )
73, 4, 5, 6islshp 35101 . . . 4  |-  ( W  e.  LMod  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  ( Base `  W
)  /\  E. v  e.  ( Base `  W
) ( ( LSpan `  W ) `  ( U  u.  { v } ) )  =  ( Base `  W
) ) ) )
82, 7syl 16 . . 3  |-  ( ph  ->  ( U  e.  H  <->  ( U  e.  S  /\  U  =/=  ( Base `  W
)  /\  E. v  e.  ( Base `  W
) ( ( LSpan `  W ) `  ( U  u.  { v } ) )  =  ( Base `  W
) ) ) )
91, 8mpbid 210 . 2  |-  ( ph  ->  ( U  e.  S  /\  U  =/=  ( Base `  W )  /\  E. v  e.  ( Base `  W ) ( (
LSpan `  W ) `  ( U  u.  { v } ) )  =  ( Base `  W
) ) )
109simp1d 1006 1  |-  ( ph  ->  U  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805    u. cun 3459   {csn 4016   ` cfv 5570   Basecbs 14716   LModclmod 17707   LSubSpclss 17773   LSpanclspn 17812  LSHypclsh 35097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-lshyp 35099
This theorem is referenced by:  lshpnel  35105  lshpnelb  35106  lshpne0  35108  lshpdisj  35109  lshpcmp  35110  lshpsmreu  35231  lshpkrlem1  35232  lshpkrlem5  35236  lshpkr  35239  dochshpncl  37508  dochshpsat  37578  lclkrlem2f  37636
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