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Theorem lshplss 29171
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 2283 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2283 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
5 lshplss.s . . . . 5  |-  S  =  ( LSubSp `  W )
6 lshplss.h . . . . 5  |-  H  =  (LSHyp `  W )
73, 4, 5, 6islshp 29169 . . . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    u. cun 3150   {csn 3640   ` cfv 5255   Basecbs 13148   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728  LSHypclsh 29165
This theorem is referenced by:  lshpnel  29173  lshpnelb  29174  lshpne0  29176  lshpdisj  29177  lshpcmp  29178  lshpsmreu  29299  lshpkrlem1  29300  lshpkrlem5  29304  lshpkr  29307  dochshpncl  31574  dochshpsat  31644  lclkrlem2f  31702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-lshyp 29167
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