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Theorem lmodacl 16939
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 16937 . 2  |-  ( W  e.  LMod  ->  F  e. 
Grp )
3 lmodacl.k . . 3  |-  K  =  ( Base `  F
)
4 lmodacl.p . . 3  |-  .+  =  ( +g  `  F )
53, 4grpcl 15544 . 2  |-  ( ( F  e.  Grp  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y
)  e.  K )
62, 5syl3an1 1246 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 960    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234  Scalarcsca 14237   Grpcgrp 15406   LModclmod 16928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-nul 4418  ax-pow 4467
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-iota 5378  df-fv 5423  df-ov 6093  df-mnd 15411  df-grp 15538  df-rng 16637  df-lmod 16930
This theorem is referenced by:  lmodcom  16971  lss1d  17022  lspsolvlem  17201  lfladdcl  32438  lshpkrlem5  32481  ldualvsdi2  32511  baerlem5blem1  35076  hgmapadd  35264
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