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Theorem lmodacl 20870
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 20867 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Grp)
3 lmodacl.k . . 3 𝐾 = (Base‘𝐹)
4 lmodacl.p . . 3 + = (+g𝐹)
53, 4grpcl 18959 . 2 ((𝐹 ∈ Grp ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
62, 5syl3an1 1164 1 ((𝑊 ∈ LMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Scalarcsca 17300  Grpcgrp 18951  LModclmod 20858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-ring 20232  df-lmod 20860
This theorem is referenced by:  lmodcom  20906  lss1d  20961  lspsolvlem  21144  lmodvslmhm  33053  imaslmod  33381  lfladdcl  39072  lshpkrlem5  39115  ldualvsdi2  39145  baerlem5blem1  41711  hgmapadd  41896
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