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Theorem lshplss 29793
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 2296 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2296 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lshplss.s . . . . 5  |-  S  =  ( LSubSp `  W )
6 lshplss.h . . . . 5  |-  H  =  (LSHyp `  W )
73, 4, 5, 6islshp 29791 . . . 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 15 . . 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 201 . 2  |-  ( ph  ->  ( U  e.  S  /\  U  =/=  ( Base `  W )  /\  E. v  e.  ( Base `  W ) ( (
LSpan `  W ) `  ( U  u.  { v } ) )  =  ( Base `  W
) ) )
109simp1d 967 1  |-  ( ph  ->  U  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    u. cun 3163   {csn 3653   ` cfv 5271   Basecbs 13164   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744  LSHypclsh 29787
This theorem is referenced by:  lshpnel  29795  lshpnelb  29796  lshpne0  29798  lshpdisj  29799  lshpcmp  29800  lshpsmreu  29921  lshpkrlem1  29922  lshpkrlem5  29926  lshpkr  29929  dochshpncl  32196  dochshpsat  32266  lclkrlem2f  32324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-lshyp 29789
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