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Theorem lssn0 18711
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 (𝑈𝑆𝑈 ≠ ∅)

Proof of Theorem lssn0
Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2609 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3 eqid 2609 . . 3 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2609 . . 3 (+g𝑊) = (+g𝑊)
5 eqid 2609 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 lssn0.s . . 3 𝑆 = (LSubSp‘𝑊)
71, 2, 3, 4, 5, 6islss 18705 . 2 (𝑈𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑎𝑈𝑏𝑈 ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑈))
87simp2bi 1069 1 (𝑈𝑆𝑈 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wne 2779  wral 2895  wss 3539  c0 3873  cfv 5790  (class class class)co 6527  Basecbs 15644  +gcplusg 15717  Scalarcsca 15720   ·𝑠 cvsca 15721  LSubSpclss 18702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-lss 18703
This theorem is referenced by:  00lss  18712  lss0cl  18717  lssne0  18721  lsssubg  18727  lbsextlem2  18929  minveclem1  22948
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