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Theorem lssn0 19705
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 2798 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2798 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3 eqid 2798 . . 3 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2798 . . 3 (+g𝑊) = (+g𝑊)
5 eqid 2798 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 lssn0.s . . 3 𝑆 = (LSubSp‘𝑊)
71, 2, 3, 4, 5, 6islss 19699 . 2 (𝑈𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑎𝑈𝑏𝑈 ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑈))
87simp2bi 1143 1 (𝑈𝑆𝑈 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  wne 2987  wral 3106  wss 3881  c0 4243  cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  Scalarcsca 16560   ·𝑠 cvsca 16561  LSubSpclss 19696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-lss 19697
This theorem is referenced by:  00lss  19706  lss0cl  19711  lssne0  19715  lsssubg  19722  lbsextlem2  19924  minveclem1  24028
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