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Theorem lmodacl 15638
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 15636 . 2  |-  ( W  e.  LMod  ->  F  e. 
Grp )
3 lmodacl.k . . 3  |-  K  =  ( Base `  F
)
4 lmodacl.p . . 3  |-  .+  =  ( +g  `  F )
53, 4grpcl 14495 . 2  |-  ( ( F  e.  Grp  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y
)  e.  K )
62, 5syl3an1 1215 1  |-  ( ( W  e.  LMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y )  e.  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   Grpcgrp 14362   LModclmod 15627
This theorem is referenced by:  lmodcom  15671  lss1d  15720  lspsolvlem  15895  lfladdcl  29261  lshpkrlem5  29304  ldualvsdi2  29334  baerlem5blem1  31899  hgmapadd  32087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-mnd 14367  df-grp 14489  df-rng 15340  df-lmod 15629
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