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Theorem lss1 17032
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 2444 . 2  |-  ( W  e.  LMod  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2444 . 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 2444 . 2  |-  ( W  e.  LMod  ->  ( +g  `  W )  =  ( +g  `  W ) )
6 eqidd 2444 . 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 3387 . . 3  |-  V  C_  V
109a1i 11 . 2  |-  ( W  e.  LMod  ->  V  C_  V )
113lmodbn0 16970 . 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 2443 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
14 eqid 2443 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
15 eqid 2443 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
163, 13, 14, 15lmodvscl 16977 . . . 4  |-  ( ( W  e.  LMod  /\  x  e.  ( Base `  (Scalar `  W ) )  /\  a  e.  V )  ->  ( x ( .s
`  W ) a )  e.  V )
17163adant3r3 1198 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  (
x ( .s `  W ) a )  e.  V )
18 simpr3 996 . . 3  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  a  e.  V  /\  b  e.  V
) )  ->  b  e.  V )
19 eqid 2443 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
203, 19lmodvacl 16974 . . 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 1218 . 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 17029 1  |-  ( W  e.  LMod  ->  V  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    C_ wss 3340   ` cfv 5430  (class class class)co 6103   Basecbs 14186   +g cplusg 14250  Scalarcsca 14253   .scvsca 14254   LModclmod 16960   LSubSpclss 17025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fv 5438  df-riota 6064  df-ov 6106  df-0g 14392  df-mnd 15427  df-grp 15557  df-lmod 16962  df-lss 17026
This theorem is referenced by:  lssuni  17033  islss3  17052  lssmre  17059  lspf  17067  lspval  17068  lmhmrnlss  17143  lidl1  17314  aspval  17411  isphld  18095  ocv1  18116  lnmfg  29447  islshpcv  32710  dochexmidlem8  35124  hdmaprnlem4N  35513
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