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Theorem lss1 17711
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 2458 . 2  |-  ( W  e.  LMod  ->  (Scalar `  W )  =  (Scalar `  W ) )
2 eqidd 2458 . 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 2458 . 2  |-  ( W  e.  LMod  ->  ( +g  `  W )  =  ( +g  `  W ) )
6 eqidd 2458 . 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 3518 . . 3  |-  V  C_  V
109a1i 11 . 2  |-  ( W  e.  LMod  ->  V  C_  V )
113lmodbn0 17648 . 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 2457 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
14 eqid 2457 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
15 eqid 2457 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
163, 13, 14, 15lmodvscl 17655 . . . 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 2457 . . . 4  |-  ( +g  `  W )  =  ( +g  `  W )
203, 19lmodvacl 17652 . . 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 17708 1  |-  ( W  e.  LMod  ->  V  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711  Scalarcsca 14714   .scvsca 14715   LModclmod 17638   LSubSpclss 17704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-riota 6258  df-ov 6299  df-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183  df-lmod 17640  df-lss 17705
This theorem is referenced by:  lssuni  17712  islss3  17731  lssmre  17738  lspf  17746  lspval  17747  lmhmrnlss  17822  lidl1  17994  aspval  18103  isphld  18815  ocv1  18836  lnmfg  31190  islshpcv  34879  dochexmidlem8  37295  hdmaprnlem4N  37684
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