Theorem lss1 13858

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 e. LMod -> V e. S )

Proof of Theorem lss1

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

Step	Hyp	Ref	Expression
1		eqidd 2194	. 2 |- ( W e. LMod -> (Scalar ` W ) = (Scalar ` W ) )
2		eqidd 2194	. 2 |- ( W e. LMod -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) ) )
3		lssss.v	. . 3 |- V = ( Base ` W )
4	3	a1i 9	. 2 |- ( W e. LMod -> V = ( Base ` W ) )
5		eqidd 2194	. 2 |- ( W e. LMod -> ( +g ` W ) = ( +g ` W ) )
6		eqidd 2194	. 2 |- ( W e. LMod -> ( .s ` W ) = ( .s ` W ) )
7		lssss.s	. . 3 |- S = ( LSubSp ` W )
8	7	a1i 9	. 2 |- ( W e. LMod -> S = ( LSubSp ` W ) )
9		ssidd 3200	. 2 |- ( W e. LMod -> V C_ V )
10		eqid 2193	. . . 4 |- ( 0g ` W ) = ( 0g ` W )
11	3, 10	lmod0vcl 13813	. . 3 |- ( W e. LMod -> ( 0g ` W ) e. V )
12		elex2 2776	. . 3 |- ( ( 0g ` W ) e. V -> E. j j e. V )
13	11, 12	syl 14	. 2 |- ( W e. LMod -> E. j j e. V )
14		simpl 109	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> W e. LMod )
15		eqid 2193	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
16		eqid 2193	. . . . 5 |- ( .s ` W ) = ( .s ` W )
17		eqid 2193	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
18	3, 15, 16, 17	lmodvscl 13801	. . . 4 |- ( ( W e. LMod /\ x e. ( Base ` (Scalar ` W ) ) /\ a e. V ) -> ( x ( .s ` W ) a ) e. V )
19	18	3adant3r3 1216	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> ( x ( .s ` W ) a ) e. V )
20		simpr3 1007	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> b e. V )
21		eqid 2193	. . . 4 |- ( +g ` W ) = ( +g ` W )
22	3, 21	lmodvacl 13798	. . 3 |- ( ( W e. LMod /\ ( x ( .s ` W ) a ) e. V /\ b e. V ) -> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. V )
23	14, 19, 20, 22	syl3anc 1249	. 2 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. V )
24		lmodgrp 13790	. 2 |- ( W e. LMod -> W e. Grp )
25	1, 2, 4, 5, 6, 8, 9, 13, 23, 24	islssmd 13855	1 |- ( W e. LMod -> V e. S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695  Scalarcsca 12698   .scvsca 12699   0gc0g 12867   Grpcgrp 13072   LModclmod 13783   LSubSpclss 13848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-lmod 13785  df-lssm 13849

This theorem is referenced by:  lssuni  13859  islss3  13875  lspf  13885  lspval  13886  lidl1  13986