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Theorem lshplss 29717
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 2436 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2436 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lshplss.s . . . . 5  |-  S  =  ( LSubSp `  W )
6 lshplss.h . . . . 5  |-  H  =  (LSHyp `  W )
73, 4, 5, 6islshp 29715 . . . 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 202 . 2  |-  ( ph  ->  ( U  e.  S  /\  U  =/=  ( Base `  W )  /\  E. v  e.  ( Base `  W ) ( (
LSpan `  W ) `  ( U  u.  { v } ) )  =  ( Base `  W
) ) )
109simp1d 969 1  |-  ( ph  ->  U  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2699    u. cun 3311   {csn 3807   ` cfv 5447   Basecbs 13462   LModclmod 15943   LSubSpclss 16001   LSpanclspn 16040  LSHypclsh 29711
This theorem is referenced by:  lshpnel  29719  lshpnelb  29720  lshpne0  29722  lshpdisj  29723  lshpcmp  29724  lshpsmreu  29845  lshpkrlem1  29846  lshpkrlem5  29850  lshpkr  29853  dochshpncl  32120  dochshpsat  32190  lclkrlem2f  32248
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 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-iota 5411  df-fun 5449  df-fv 5455  df-lshyp 29713
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