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Theorem lmodacl 19621
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 19619 . 2 (𝑊 ∈ LMod → 𝐹 ∈ Grp)
3 lmodacl.k . . 3 𝐾 = (Base‘𝐹)
4 lmodacl.p . . 3 + = (+g𝐹)
53, 4grpcl 18090 . 2 ((𝐹 ∈ Grp ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
62, 5syl3an1 1160 1 ((𝑊 ∈ LMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  cfv 6328  (class class class)co 7130  Basecbs 16462  +gcplusg 16544  Scalarcsca 16547  Grpcgrp 18082  LModclmod 19610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-iota 6287  df-fv 6336  df-ov 7133  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-grp 18085  df-ring 19278  df-lmod 19612
This theorem is referenced by:  lmodcom  19656  lss1d  19711  lspsolvlem  19890  lmodvslmhm  30696  imaslmod  30930  lfladdcl  36249  lshpkrlem5  36292  ldualvsdi2  36322  baerlem5blem1  38887  hgmapadd  39072
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