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Theorem lss1 20954
Description: The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lssss.v 𝑉 = (Base‘𝑊)
lssss.s 𝑆 = (LSubSp‘𝑊)
Assertion
Ref Expression
lss1 (𝑊 ∈ LMod → 𝑉𝑆)

Proof of Theorem lss1
Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2736 . 2 (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊))
2 eqidd 2736 . 2 (𝑊 ∈ LMod → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
3 lssss.v . . 3 𝑉 = (Base‘𝑊)
43a1i 11 . 2 (𝑊 ∈ LMod → 𝑉 = (Base‘𝑊))
5 eqidd 2736 . 2 (𝑊 ∈ LMod → (+g𝑊) = (+g𝑊))
6 eqidd 2736 . 2 (𝑊 ∈ LMod → ( ·𝑠𝑊) = ( ·𝑠𝑊))
7 lssss.s . . 3 𝑆 = (LSubSp‘𝑊)
87a1i 11 . 2 (𝑊 ∈ LMod → 𝑆 = (LSubSp‘𝑊))
9 ssidd 4019 . 2 (𝑊 ∈ LMod → 𝑉𝑉)
103lmodbn0 20886 . 2 (𝑊 ∈ LMod → 𝑉 ≠ ∅)
11 simpl 482 . . 3 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → 𝑊 ∈ LMod)
12 eqid 2735 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
13 eqid 2735 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
14 eqid 2735 . . . . 5 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
153, 12, 13, 14lmodvscl 20893 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉) → (𝑥( ·𝑠𝑊)𝑎) ∈ 𝑉)
16153adant3r3 1183 . . 3 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → (𝑥( ·𝑠𝑊)𝑎) ∈ 𝑉)
17 simpr3 1195 . . 3 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → 𝑏𝑉)
18 eqid 2735 . . . 4 (+g𝑊) = (+g𝑊)
193, 18lmodvacl 20890 . . 3 ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠𝑊)𝑎) ∈ 𝑉𝑏𝑉) → ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑉)
2011, 16, 17, 19syl3anc 1370 . 2 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑉)
211, 2, 4, 5, 6, 8, 9, 10, 20islssd 20951 1 (𝑊 ∈ LMod → 𝑉𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Scalarcsca 17301   ·𝑠 cvsca 17302  LModclmod 20875  LSubSpclss 20947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-lmod 20877  df-lss 20948
This theorem is referenced by:  lssuni  20955  islss3  20975  lssmre  20982  lspf  20990  lspval  20991  lmhmrnlss  21067  lidl1ALT  21259  isphld  21690  ocv1  21715  aspval  21911  islshpcv  39035  dochexmidlem8  41450  hdmaprnlem4N  41836  lnmfg  43071
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