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Theorem lssn0 19145
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 2806 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
2 eqid 2806 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
3 eqid 2806 . . 3 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2806 . . 3 (+g𝑊) = (+g𝑊)
5 eqid 2806 . . 3 ( ·𝑠𝑊) = ( ·𝑠𝑊)
6 lssn0.s . . 3 𝑆 = (LSubSp‘𝑊)
71, 2, 3, 4, 5, 6islss 19139 . 2 (𝑈𝑆 ↔ (𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑎𝑈𝑏𝑈 ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑈))
87simp2bi 1169 1 (𝑈𝑆𝑈 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  wne 2978  wral 3096  wss 3769  c0 4116  cfv 6101  (class class class)co 6874  Basecbs 16068  +gcplusg 16153  Scalarcsca 16156   ·𝑠 cvsca 16157  LSubSpclss 19136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6877  df-lss 19137
This theorem is referenced by:  00lss  19146  lss0cl  19151  lssne0  19155  lsssubg  19164  lbsextlem2  19368  minveclem1  23407
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