Theorem lss1 19710

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 → 𝑉 ∈ 𝑆)

Proof of Theorem lss1

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2822	. 2 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) = (Scalar‘𝑊))
2		eqidd 2822	. 2 ⊢ (𝑊 ∈ LMod → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
3		lssss.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
4	3	a1i 11	. 2 ⊢ (𝑊 ∈ LMod → 𝑉 = (Base‘𝑊))
5		eqidd 2822	. 2 ⊢ (𝑊 ∈ LMod → (+g‘𝑊) = (+g‘𝑊))
6		eqidd 2822	. 2 ⊢ (𝑊 ∈ LMod → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊))
7		lssss.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑊)
8	7	a1i 11	. 2 ⊢ (𝑊 ∈ LMod → 𝑆 = (LSubSp‘𝑊))
9		ssidd 3990	. 2 ⊢ (𝑊 ∈ LMod → 𝑉 ⊆ 𝑉)
10	3	lmodbn0 19644	. 2 ⊢ (𝑊 ∈ LMod → 𝑉 ≠ ∅)
11		simpl 485	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑊 ∈ LMod)
12		eqid 2821	. . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
13		eqid 2821	. . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
14		eqid 2821	. . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
15	3, 12, 13, 14	lmodvscl 19651	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉) → (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉)
16	15	3adant3r3 1180	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉)
17		simpr3 1192	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
18		eqid 2821	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
19	3, 18	lmodvacl 19648	. . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑥( ·𝑠 ‘𝑊)𝑎) ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑉)
20	11, 16, 17, 19	syl3anc 1367	. 2 ⊢ ((𝑊 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑥( ·𝑠 ‘𝑊)𝑎)(+g‘𝑊)𝑏) ∈ 𝑉)
21	1, 2, 4, 5, 6, 8, 9, 10, 20	islssd 19707	1 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  +_gcplusg 16565  Scalarcsca 16568   ·_𝑠 cvsca 16569  LModclmod 19634  LSubSpclss 19703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-riota 7114  df-ov 7159  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-lmod 19636  df-lss 19704

This theorem is referenced by:  lssuni  19711  islss3  19731  lssmre  19738  lspf  19746  lspval  19747  lmhmrnlss  19822  lidl1  19993  aspval  20102  isphld  20798  ocv1  20823  islshpcv  36204  dochexmidlem8  38618  hdmaprnlem4N  39004  lnmfg  39702