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Theorem lmodacl 18922
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 18920 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Grp)
3 lmodacl.k . . 3 𝐾 = (Base‘𝐹)
4 lmodacl.p . . 3 + = (+g𝐹)
53, 4grpcl 17477 . 2 ((𝐹 ∈ Grp ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
62, 5syl3an1 1399 1 ((𝑊 ∈ LMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1054   = wceq 1523  wcel 2030  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Scalarcsca 15991  Grpcgrp 17469  LModclmod 18911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-ring 18595  df-lmod 18913
This theorem is referenced by:  lmodcom  18957  lss1d  19011  lspsolvlem  19190  lfladdcl  34676  lshpkrlem5  34719  ldualvsdi2  34749  baerlem5blem1  37315  hgmapadd  37503
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