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Theorem lssn0 19150
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 2770 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2770 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3 eqid 2770 . . 3 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2770 . . 3 (+g𝑊) = (+g𝑊)
5 eqid 2770 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 lssn0.s . . 3 𝑆 = (LSubSp‘𝑊)
71, 2, 3, 4, 5, 6islss 19144 . 2 (𝑈𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑎𝑈𝑏𝑈 ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑈))
87simp2bi 1139 1 (𝑈𝑆𝑈 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  wne 2942  wral 3060  wss 3721  c0 4061  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  Scalarcsca 16151   ·𝑠 cvsca 16152  LSubSpclss 19141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-lss 19142
This theorem is referenced by:  00lss  19151  lss0cl  19156  lssne0  19160  lsssubg  19169  lbsextlem2  19373  minveclem1  23413
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