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Theorem lmodacl 19943
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 19941 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Grp)
3 lmodacl.k . . 3 𝐾 = (Base‘𝐹)
4 lmodacl.p . . 3 + = (+g𝐹)
53, 4grpcl 18406 . 2 ((𝐹 ∈ Grp ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
62, 5syl3an1 1165 1 ((𝑊 ∈ LMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2112  cfv 6401  (class class class)co 7235  Basecbs 16793  +gcplusg 16835  Scalarcsca 16838  Grpcgrp 18398  LModclmod 19932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-nul 5216
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4255  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6359  df-fv 6409  df-ov 7238  df-mgm 18147  df-sgrp 18196  df-mnd 18207  df-grp 18401  df-ring 19597  df-lmod 19934
This theorem is referenced by:  lmodcom  19978  lss1d  20033  lspsolvlem  20212  lmodvslmhm  31061  imaslmod  31299  lfladdcl  36859  lshpkrlem5  36902  ldualvsdi2  36932  baerlem5blem1  39497  hgmapadd  39682
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