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Theorem lssn0 19337
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 2778 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2778 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3 eqid 2778 . . 3 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2778 . . 3 (+g𝑊) = (+g𝑊)
5 eqid 2778 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 lssn0.s . . 3 𝑆 = (LSubSp‘𝑊)
71, 2, 3, 4, 5, 6islss 19331 . 2 (𝑈𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑎𝑈𝑏𝑈 ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑈))
87simp2bi 1137 1 (𝑈𝑆𝑈 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  wne 2969  wral 3090  wss 3792  c0 4141  cfv 6137  (class class class)co 6924  Basecbs 16259  +gcplusg 16342  Scalarcsca 16345   ·𝑠 cvsca 16346  LSubSpclss 19328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927  df-lss 19329
This theorem is referenced by:  00lss  19338  lss0cl  19343  lssne0  19347  lsssubg  19356  lbsextlem2  19560  minveclem1  23634
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