Theorem lss1 14748

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 2239	. 2 |- ( W e. LMod -> (Scalar ` W ) = (Scalar ` W ) )
2		eqidd 2239	. 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 2239	. 2 |- ( W e. LMod -> ( +g ` W ) = ( +g ` W ) )
6		eqidd 2239	. 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 3269	. 2 |- ( W e. LMod -> V C_ V )
10		eqid 2238	. . . 4 |- ( 0g ` W ) = ( 0g ` W )
11	3, 10	lmod0vcl 14703	. . 3 |- ( W e. LMod -> ( 0g ` W ) e. V )
12		elex2 2838	. . 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 2238	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
16		eqid 2238	. . . . 5 |- ( .s ` W ) = ( .s ` W )
17		eqid 2238	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
18	3, 15, 16, 17	lmodvscl 14690	. . . 4 |- ( ( W e. LMod /\ x e. ( Base ` (Scalar ` W ) ) /\ a e. V ) -> ( x ( .s ` W ) a ) e. V )
19	18	3adant3r3 1245	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> ( x ( .s ` W ) a ) e. V )
20		simpr3 1036	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> b e. V )
21		eqid 2238	. . . 4 |- ( +g ` W ) = ( +g ` W )
22	3, 21	lmodvacl 14687	. . 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 1278	. 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 14679	. 2 |- ( W e. LMod -> W e. Grp )
25	1, 2, 4, 5, 6, 8, 9, 13, 23, 24	islssmd 14745	1 |- ( W e. LMod -> V e. S )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   ` cfv 5377  (class class class)co 6085   Basecbs 13401   +g cplusg 13480  Scalarcsca 13483   .scvsca 13484   0gc0g 13659   Grpcgrp 13854   LModclmod 14672   LSubSpclss 14738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-5 9368  df-6 9369  df-ndx 13404  df-slot 13405  df-base 13407  df-plusg 13493  df-mulr 13494  df-sca 13496  df-vsca 13497  df-0g 13661  df-mgm 13725  df-sgrp 13766  df-mnd 13779  df-grp 13857  df-lmod 14674  df-lssm 14739

This theorem is used by:  lssuni  14749  islss3  14765  lspf  14775  lspval  14776  lidl1  14876  aspval  15064  asplss  15065  aspsubrg  15067