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Theorem lmodacl 18643
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 + = (+g𝐹)
Assertion
Ref Expression
lmodacl ((𝑊 ∈ LMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)

Proof of Theorem lmodacl
StepHypRef Expression
1 lmodacl.f . . 3 𝐹 = (Scalar‘𝑊)
21lmodfgrp 18641 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Grp)
3 lmodacl.k . . 3 𝐾 = (Base‘𝐹)
4 lmodacl.p . . 3 + = (+g𝐹)
53, 4grpcl 17199 . 2 ((𝐹 ∈ Grp ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
62, 5syl3an1 1350 1 ((𝑊 ∈ LMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1030   = wceq 1474  wcel 1976  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  Scalarcsca 15717  Grpcgrp 17191  LModclmod 18632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-grp 17194  df-ring 18318  df-lmod 18634
This theorem is referenced by:  lmodcom  18678  lss1d  18730  lspsolvlem  18909  lfladdcl  33179  lshpkrlem5  33222  ldualvsdi2  33252  baerlem5blem1  35819  hgmapadd  36007
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