Theorem lss1 13454

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 2178	. 2 |- ( W e. LMod -> (Scalar ` W ) = (Scalar ` W ) )
2		eqidd 2178	. 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 2178	. 2 |- ( W e. LMod -> ( +g ` W ) = ( +g ` W ) )
6		eqidd 2178	. 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 3178	. 2 |- ( W e. LMod -> V C_ V )
10		eqid 2177	. . . 4 |- ( 0g ` W ) = ( 0g ` W )
11	3, 10	lmod0vcl 13412	. . 3 |- ( W e. LMod -> ( 0g ` W ) e. V )
12		elex2 2755	. . 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 2177	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
16		eqid 2177	. . . . 5 |- ( .s ` W ) = ( .s ` W )
17		eqid 2177	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
18	3, 15, 16, 17	lmodvscl 13400	. . . 4 |- ( ( W e. LMod /\ x e. ( Base ` (Scalar ` W ) ) /\ a e. V ) -> ( x ( .s ` W ) a ) e. V )
19	18	3adant3r3 1214	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> ( x ( .s ` W ) a ) e. V )
20		simpr3 1005	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> b e. V )
21		eqid 2177	. . . 4 |- ( +g ` W ) = ( +g ` W )
22	3, 21	lmodvacl 13397	. . 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 1238	. 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 13389	. 2 |- ( W e. LMod -> W e. Grp )
25	1, 2, 4, 5, 6, 8, 9, 13, 23, 24	islssmd 13451	1 |- ( W e. LMod -> V e. S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538  Scalarcsca 12541   .scvsca 12542   0gc0g 12710   Grpcgrp 12882   LModclmod 13382   LSubSpclss 13447

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-mulr 12552  df-sca 12554  df-vsca 12555  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-lmod 13384  df-lssm 13448

This theorem is referenced by:  lssuni  13455  islss3  13471  lspf  13481  lspval  13482