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Theorem slmdacl 32910
Description: Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdacl.f 𝐹 = (Scalar‘𝑊)
slmdacl.k 𝐾 = (Base‘𝐹)
slmdacl.p + = (+g𝐹)
Assertion
Ref Expression
slmdacl ((𝑊 ∈ SLMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)

Proof of Theorem slmdacl
StepHypRef Expression
1 slmdacl.f . . . 4 𝐹 = (Scalar‘𝑊)
21slmdsrg 32908 . . 3 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 srgmnd 20123 . . 3 (𝐹 ∈ SRing → 𝐹 ∈ Mnd)
42, 3syl 17 . 2 (𝑊 ∈ SLMod → 𝐹 ∈ Mnd)
5 slmdacl.k . . 3 𝐾 = (Base‘𝐹)
6 slmdacl.p . . 3 + = (+g𝐹)
75, 6mndcl 18695 . 2 ((𝐹 ∈ Mnd ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
84, 7syl3an1 1161 1 ((𝑊 ∈ SLMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1534  wcel 2099  cfv 6542  (class class class)co 7414  Basecbs 17173  +gcplusg 17226  Scalarcsca 17229  Mndcmnd 18687  SRingcsrg 20119  SLModcslmd 32901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-nul 5300
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-cmn 19730  df-srg 20120  df-slmd 32902
This theorem is referenced by: (None)
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