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Theorem lssn0 20817
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 = (LSubSpβ€˜π‘Š)
Assertion
Ref Expression
lssn0 (π‘ˆ ∈ 𝑆 β†’ π‘ˆ β‰  βˆ…)

Proof of Theorem lssn0
Dummy variables π‘Ž 𝑏 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . . 3 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
2 eqid 2728 . . 3 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
3 eqid 2728 . . 3 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
4 eqid 2728 . . 3 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
5 eqid 2728 . . 3 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
6 lssn0.s . . 3 𝑆 = (LSubSpβ€˜π‘Š)
71, 2, 3, 4, 5, 6islss 20811 . 2 (π‘ˆ ∈ 𝑆 ↔ (π‘ˆ βŠ† (Baseβ€˜π‘Š) ∧ π‘ˆ β‰  βˆ… ∧ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘Ž ∈ π‘ˆ βˆ€π‘ ∈ π‘ˆ ((π‘₯( ·𝑠 β€˜π‘Š)π‘Ž)(+gβ€˜π‘Š)𝑏) ∈ π‘ˆ))
87simp2bi 1144 1 (π‘ˆ ∈ 𝑆 β†’ π‘ˆ β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099   β‰  wne 2936  βˆ€wral 3057   βŠ† wss 3945  βˆ…c0 4318  β€˜cfv 6542  (class class class)co 7414  Basecbs 17173  +gcplusg 17226  Scalarcsca 17229   ·𝑠 cvsca 17230  LSubSpclss 20808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-lss 20809
This theorem is referenced by:  00lss  20818  lss0cl  20824  lssne0  20828  lsssubg  20834  lbsextlem2  21040  minveclem1  25345
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