MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmodacl Structured version   Unicode version

Theorem lmodacl 17391
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 17389 . 2  |-  ( W  e.  LMod  ->  F  e. 
Grp )
3 lmodacl.k . . 3  |-  K  =  ( Base `  F
)
4 lmodacl.p . . 3  |-  .+  =  ( +g  `  F )
53, 4grpcl 15932 . 2  |-  ( ( F  e.  Grp  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  .+  Y
)  e.  K )
62, 5syl3an1 1260 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 972    = wceq 1381    e. wcel 1802   ` cfv 5574  (class class class)co 6277   Basecbs 14504   +g cplusg 14569  Scalarcsca 14572   Grpcgrp 15922   LModclmod 17380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-nul 4562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-iota 5537  df-fv 5582  df-ov 6280  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-ring 17068  df-lmod 17382
This theorem is referenced by:  lmodcom  17424  lss1d  17477  lspsolvlem  17656  lfladdcl  34498  lshpkrlem5  34541  ldualvsdi2  34571  baerlem5blem1  37138  hgmapadd  37326
  Copyright terms: Public domain W3C validator