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Theorem lss1 16016
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  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2438 . 2  |-  ( W  e.  LMod  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2438 . 2  |-  ( W  e.  LMod  ->  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
) )
3 lssss.v . . 3  |-  V  =  ( Base `  W
)
43a1i 11 . 2  |-  ( W  e.  LMod  ->  V  =  ( Base `  W
) )
5 eqidd 2438 . 2  |-  ( W  e.  LMod  ->  ( +g  `  W )  =  ( +g  `  W ) )
6 eqidd 2438 . 2  |-  ( W  e.  LMod  ->  ( .s
`  W )  =  ( .s `  W
) )
7 lssss.s . . 3  |-  S  =  ( LSubSp `  W )
87a1i 11 . 2  |-  ( W  e.  LMod  ->  S  =  ( LSubSp `  W )
)
9 ssid 3368 . . 3  |-  V  C_  V
109a1i 11 . 2  |-  ( W  e.  LMod  ->  V  C_  V )
113lmodbn0 15961 . 2  |-  ( W  e.  LMod  ->  V  =/=  (/) )
12 simpl 445 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  W  e.  LMod )
13 eqid 2437 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
14 eqid 2437 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
15 eqid 2437 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
163, 13, 14, 15lmodvscl 15968 . . . 4  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  V )  ->  ( x ( .s
`  W ) a )  e.  V )
17163adant3r3 1165 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  (
x ( .s `  W ) a )  e.  V )
18 simpr3 966 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  b  e.  V )
19 eqid 2437 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
203, 19lmodvacl 15965 . . 3  |-  ( ( W  e.  LMod  /\  (
x ( .s `  W ) a )  e.  V  /\  b  e.  V )  ->  (
( x ( .s
`  W ) a ) ( +g  `  W
) b )  e.  V )
2112, 17, 18, 20syl3anc 1185 . 2  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  (
( x ( .s
`  W ) a ) ( +g  `  W
) b )  e.  V )
221, 2, 4, 5, 6, 8, 10, 11, 21islssd 16013 1  |-  ( W  e.  LMod  ->  V  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3321   ` cfv 5455  (class class class)co 6082   Basecbs 13470   +g cplusg 13530  Scalarcsca 13533   .scvsca 13534   LModclmod 15951   LSubSpclss 16009
This theorem is referenced by:  lssuni  16017  islss3  16036  lssmre  16043  lspf  16051  lspval  16052  lmhmrnlss  16127  lidl1  16292  aspval  16388  isphld  16886  ocv1  16907  lnmfg  27158  islshpcv  29852  dochexmidlem8  32266  hdmaprnlem4N  32655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-riota 6550  df-0g 13728  df-mnd 14691  df-grp 14813  df-lmod 15953  df-lss 16010
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