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Theorem lss1 20936
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 2738 . 2 (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊))
2 eqidd 2738 . 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 2738 . 2 (𝑊 ∈ LMod → (+g𝑊) = (+g𝑊))
6 eqidd 2738 . 2 (𝑊 ∈ LMod → ( ·𝑠𝑊) = ( ·𝑠𝑊))
7 lssss.s . . 3 𝑆 = (LSubSp‘𝑊)
87a1i 11 . 2 (𝑊 ∈ LMod → 𝑆 = (LSubSp‘𝑊))
9 ssidd 4007 . 2 (𝑊 ∈ LMod → 𝑉𝑉)
103lmodbn0 20869 . 2 (𝑊 ∈ LMod → 𝑉 ≠ ∅)
11 simpl 482 . . 3 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → 𝑊 ∈ LMod)
12 eqid 2737 . . . . 5 (Scalar‘𝑊) = (Scalar‘𝑊)
13 eqid 2737 . . . . 5 ( ·𝑠𝑊) = ( ·𝑠𝑊)
14 eqid 2737 . . . . 5 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
153, 12, 13, 14lmodvscl 20876 . . . 4 ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉) → (𝑥( ·𝑠𝑊)𝑎) ∈ 𝑉)
16153adant3r3 1185 . . 3 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → (𝑥( ·𝑠𝑊)𝑎) ∈ 𝑉)
17 simpr3 1197 . . 3 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → 𝑏𝑉)
18 eqid 2737 . . . 4 (+g𝑊) = (+g𝑊)
193, 18lmodvacl 20873 . . 3 ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠𝑊)𝑎) ∈ 𝑉𝑏𝑉) → ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑉)
2011, 16, 17, 19syl3anc 1373 . 2 ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎𝑉𝑏𝑉)) → ((𝑥( ·𝑠𝑊)𝑎)(+g𝑊)𝑏) ∈ 𝑉)
211, 2, 4, 5, 6, 8, 9, 10, 20islssd 20933 1 (𝑊 ∈ LMod → 𝑉𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Scalarcsca 17300   ·𝑠 cvsca 17301  LModclmod 20858  LSubSpclss 20929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-lmod 20860  df-lss 20930
This theorem is referenced by:  lssuni  20937  islss3  20957  lssmre  20964  lspf  20972  lspval  20973  lmhmrnlss  21049  lidl1ALT  21241  isphld  21672  ocv1  21697  aspval  21893  islshpcv  39054  dochexmidlem8  41469  hdmaprnlem4N  41855  lnmfg  43094
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