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Theorem lss1 17397
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 2468 . 2  |-  ( W  e.  LMod  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2468 . 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 2468 . 2  |-  ( W  e.  LMod  ->  ( +g  `  W )  =  ( +g  `  W ) )
6 eqidd 2468 . 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 3523 . . 3  |-  V  C_  V
109a1i 11 . 2  |-  ( W  e.  LMod  ->  V  C_  V )
113lmodbn0 17334 . 2  |-  ( W  e.  LMod  ->  V  =/=  (/) )
12 simpl 457 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  W  e.  LMod )
13 eqid 2467 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
14 eqid 2467 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
15 eqid 2467 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
163, 13, 14, 15lmodvscl 17341 . . . 4  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  V )  ->  ( x ( .s
`  W ) a )  e.  V )
17163adant3r3 1207 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  (
x ( .s `  W ) a )  e.  V )
18 simpr3 1004 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  b  e.  V )
19 eqid 2467 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
203, 19lmodvacl 17338 . . 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 1228 . 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 17394 1  |-  ( W  e.  LMod  ->  V  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   ` cfv 5588  (class class class)co 6285   Basecbs 14493   +g cplusg 14558  Scalarcsca 14561   .scvsca 14562   LModclmod 17324   LSubSpclss 17390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-riota 6246  df-ov 6288  df-0g 14700  df-mnd 15735  df-grp 15871  df-lmod 17326  df-lss 17391
This theorem is referenced by:  lssuni  17398  islss3  17417  lssmre  17424  lspf  17432  lspval  17433  lmhmrnlss  17508  lidl1  17679  aspval  17788  isphld  18496  ocv1  18517  lnmfg  30859  islshpcv  34067  dochexmidlem8  36481  hdmaprnlem4N  36870
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