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Theorem lmodacl 17721
Description: Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
lmodacl.f  |-  F  =  (Scalar `  W )
lmodacl.k  |-  K  =  ( Base `  F
)
lmodacl.p  |-  .+  =  ( +g  `  F )
Assertion
Ref Expression
lmodacl  |-  ( ( W  e.  LMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y )  e.  K )

Proof of Theorem lmodacl
StepHypRef Expression
1 lmodacl.f . . 3  |-  F  =  (Scalar `  W )
21lmodfgrp 17719 . 2  |-  ( W  e.  LMod  ->  F  e. 
Grp )
3 lmodacl.k . . 3  |-  K  =  ( Base `  F
)
4 lmodacl.p . . 3  |-  .+  =  ( +g  `  F )
53, 4grpcl 16265 . 2  |-  ( ( F  e.  Grp  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y
)  e.  K )
62, 5syl3an1 1259 1  |-  ( ( W  e.  LMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y )  e.  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787  Scalarcsca 14790   Grpcgrp 16255   LModclmod 17710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-ring 17398  df-lmod 17712
This theorem is referenced by:  lmodcom  17754  lss1d  17807  lspsolvlem  17986  lfladdcl  35212  lshpkrlem5  35255  ldualvsdi2  35285  baerlem5blem1  37852  hgmapadd  38040
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