Theorem lss1 14375

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 2232	. 2 |- ( W e. LMod -> (Scalar ` W ) = (Scalar ` W ) )
2		eqidd 2232	. 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 2232	. 2 |- ( W e. LMod -> ( +g ` W ) = ( +g ` W ) )
6		eqidd 2232	. 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 3248	. 2 |- ( W e. LMod -> V C_ V )
10		eqid 2231	. . . 4 |- ( 0g ` W ) = ( 0g ` W )
11	3, 10	lmod0vcl 14330	. . 3 |- ( W e. LMod -> ( 0g ` W ) e. V )
12		elex2 2819	. . 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 2231	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
16		eqid 2231	. . . . 5 |- ( .s ` W ) = ( .s ` W )
17		eqid 2231	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
18	3, 15, 16, 17	lmodvscl 14318	. . . 4 |- ( ( W e. LMod /\ x e. ( Base ` (Scalar ` W ) ) /\ a e. V ) -> ( x ( .s ` W ) a ) e. V )
19	18	3adant3r3 1240	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> ( x ( .s ` W ) a ) e. V )
20		simpr3 1031	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> b e. V )
21		eqid 2231	. . . 4 |- ( +g ` W ) = ( +g ` W )
22	3, 21	lmodvacl 14315	. . 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 1273	. 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 14307	. 2 |- ( W e. LMod -> W e. Grp )
25	1, 2, 4, 5, 6, 8, 9, 13, 23, 24	islssmd 14372	1 |- ( W e. LMod -> V e. S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159  Scalarcsca 13162   .scvsca 13163   0gc0g 13338   Grpcgrp 13582   LModclmod 14300   LSubSpclss 14365

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-mulr 13173  df-sca 13175  df-vsca 13176  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-lmod 14302  df-lssm 14366

This theorem is referenced by:  lssuni  14376  islss3  14392  lspf  14402  lspval  14403  lidl1  14503