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Theorem lmodacl 17077
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 17075 . 2  |-  ( W  e.  LMod  ->  F  e. 
Grp )
3 lmodacl.k . . 3  |-  K  =  ( Base `  F
)
4 lmodacl.p . . 3  |-  .+  =  ( +g  `  F )
53, 4grpcl 15665 . 2  |-  ( ( F  e.  Grp  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y
)  e.  K )
62, 5syl3an1 1252 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 965    = wceq 1370    e. wcel 1758   ` cfv 5521  (class class class)co 6195   Basecbs 14287   +g cplusg 14352  Scalarcsca 14355   Grpcgrp 15524   LModclmod 17066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-nul 4524  ax-pow 4573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-iota 5484  df-fv 5529  df-ov 6198  df-mnd 15529  df-grp 15659  df-rng 16765  df-lmod 17068
This theorem is referenced by:  lmodcom  17109  lss1d  17162  lspsolvlem  17341  lfladdcl  33035  lshpkrlem5  33078  ldualvsdi2  33108  baerlem5blem1  35673  hgmapadd  35861
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