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Theorem lss1 15696
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 2284 . 2  |-  ( W  e.  LMod  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2284 . 2  |-  ( W  e.  LMod  ->  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
) )
3 lssss.v . . 3  |-  V  =  ( Base `  W
)
43a1i 10 . 2  |-  ( W  e.  LMod  ->  V  =  ( Base `  W
) )
5 eqidd 2284 . 2  |-  ( W  e.  LMod  ->  ( +g  `  W )  =  ( +g  `  W ) )
6 eqidd 2284 . 2  |-  ( W  e.  LMod  ->  ( .s
`  W )  =  ( .s `  W
) )
7 lssss.s . . 3  |-  S  =  ( LSubSp `  W )
87a1i 10 . 2  |-  ( W  e.  LMod  ->  S  =  ( LSubSp `  W )
)
9 ssid 3197 . . 3  |-  V  C_  V
109a1i 10 . 2  |-  ( W  e.  LMod  ->  V  C_  V )
113lmodbn0 15637 . 2  |-  ( W  e.  LMod  ->  V  =/=  (/) )
12 simpl 443 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  W  e.  LMod )
13 eqid 2283 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
14 eqid 2283 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
15 eqid 2283 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
163, 13, 14, 15lmodvscl 15644 . . . 4  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  V )  ->  ( x ( .s
`  W ) a )  e.  V )
17163adant3r3 1162 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  (
x ( .s `  W ) a )  e.  V )
18 simpr3 963 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  b  e.  V )
19 eqid 2283 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
203, 19lmodvacl 15641 . . 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 1182 . 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 15693 1  |-  ( W  e.  LMod  ->  V  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   LModclmod 15627   LSubSpclss 15689
This theorem is referenced by:  lssuni  15697  islss3  15716  lssmre  15723  lspf  15731  lspval  15732  lmhmrnlss  15807  lidl1  15972  aspval  16068  isphld  16558  ocv1  16579  lnmfg  27180  islshpcv  29243  dochexmidlem8  31657  hdmaprnlem4N  32046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-lmod 15629  df-lss 15690
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