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Theorem lssn0 15714
Description: A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
Hypothesis
Ref Expression
lssn0.s  |-  S  =  ( LSubSp `  W )
Assertion
Ref Expression
lssn0  |-  ( U  e.  S  ->  U  =/=  (/) )

Proof of Theorem lssn0
Dummy variables  a 
b  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
2 eqid 2296 . . 3  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3 eqid 2296 . . 3  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2296 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
5 eqid 2296 . . 3  |-  ( .s
`  W )  =  ( .s `  W
)
6 lssn0.s . . 3  |-  S  =  ( LSubSp `  W )
71, 2, 3, 4, 5, 6islss 15708 . 2  |-  ( U  e.  S  <->  ( U  C_  ( Base `  W
)  /\  U  =/=  (/) 
/\  A. x  e.  (
Base `  (Scalar `  W
) ) A. a  e.  U  A. b  e.  U  ( (
x ( .s `  W ) a ) ( +g  `  W
) b )  e.  U ) )
87simp2bi 971 1  |-  ( U  e.  S  ->  U  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    C_ wss 3165   (/)c0 3468   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   LSubSpclss 15705
This theorem is referenced by:  00lss  15715  lss0cl  15720  lssne0  15724  lsssubg  15730  lbsextlem2  15928  minveclem1  18804
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-13 1698  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-pow 4204  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-pw 3640  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-ov 5877  df-lss 15706
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