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Theorem lss1 18760
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 2611 . 2 (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊))
2 eqidd 2611 . 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 2611 . 2 (𝑊 ∈ LMod → (+g𝑊) = (+g𝑊))
6 eqidd 2611 . 2 (𝑊 ∈ LMod → ( ·𝑠𝑊) = ( ·𝑠𝑊))
7 lssss.s . . 3 𝑆 = (LSubSp‘𝑊)
87a1i 11 . 2 (𝑊 ∈ LMod → 𝑆 = (LSubSp‘𝑊))
9 ssid 3587 . . 3 𝑉𝑉
109a1i 11 . 2 (𝑊 ∈ LMod → 𝑉𝑉)
113lmodbn0 18696 . 2 (𝑊 ∈ LMod → 𝑉 ≠ ∅)
12 simpl 472 . . 3 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → 𝑊 ∈ LMod)
13 eqid 2610 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
14 eqid 2610 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
15 eqid 2610 . . . . 5 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
163, 13, 14, 15lmodvscl 18703 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉) → (𝑥( ·𝑠𝑊)𝑎) ∈ 𝑉)
17163adant3r3 1268 . . 3 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → (𝑥( ·𝑠𝑊)𝑎) ∈ 𝑉)
18 simpr3 1062 . . 3 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → 𝑏𝑉)
19 eqid 2610 . . . 4 (+g𝑊) = (+g𝑊)
203, 19lmodvacl 18700 . . 3 ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠𝑊)𝑎) ∈ 𝑉𝑏𝑉) → ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑉)
2112, 17, 18, 20syl3anc 1318 . 2 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑉)
221, 2, 4, 5, 6, 8, 10, 11, 21islssd 18757 1 (𝑊 ∈ LMod → 𝑉𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wss 3540  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Scalarcsca 15771   ·𝑠 cvsca 15772  LModclmod 18686  LSubSpclss 18753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-lmod 18688  df-lss 18754
This theorem is referenced by:  lssuni  18761  islss3  18780  lssmre  18787  lspf  18795  lspval  18796  lmhmrnlss  18871  lidl1  19041  aspval  19149  isphld  19818  ocv1  19842  islshpcv  33358  dochexmidlem8  35774  hdmaprnlem4N  36163  lnmfg  36670
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