Theorem lss1 14124

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 2206	. 2 |- ( W e. LMod -> (Scalar ` W ) = (Scalar ` W ) )
2		eqidd 2206	. 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 2206	. 2 |- ( W e. LMod -> ( +g ` W ) = ( +g ` W ) )
6		eqidd 2206	. 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 3214	. 2 |- ( W e. LMod -> V C_ V )
10		eqid 2205	. . . 4 |- ( 0g ` W ) = ( 0g ` W )
11	3, 10	lmod0vcl 14079	. . 3 |- ( W e. LMod -> ( 0g ` W ) e. V )
12		elex2 2788	. . 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 2205	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
16		eqid 2205	. . . . 5 |- ( .s ` W ) = ( .s ` W )
17		eqid 2205	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
18	3, 15, 16, 17	lmodvscl 14067	. . . 4 |- ( ( W e. LMod /\ x e. ( Base ` (Scalar ` W ) ) /\ a e. V ) -> ( x ( .s ` W ) a ) e. V )
19	18	3adant3r3 1217	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> ( x ( .s ` W ) a ) e. V )
20		simpr3 1008	. . 3 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ a e. V /\ b e. V ) ) -> b e. V )
21		eqid 2205	. . . 4 |- ( +g ` W ) = ( +g ` W )
22	3, 21	lmodvacl 14064	. . 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 1250	. 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 14056	. 2 |- ( W e. LMod -> W e. Grp )
25	1, 2, 4, 5, 6, 8, 9, 13, 23, 24	islssmd 14121	1 |- ( W e. LMod -> V e. S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   E.wex 1515   e. wcel 2176   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909  Scalarcsca 12912   .scvsca 12913   0gc0g 13088   Grpcgrp 13332   LModclmod 14049   LSubSpclss 14114

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-mulr 12923  df-sca 12925  df-vsca 12926  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-lmod 14051  df-lssm 14115

This theorem is referenced by:  lssuni  14125  islss3  14141  lspf  14151  lspval  14152  lidl1  14252