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Theorem slmdacl 29736
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 29734 . . 3 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 srgmnd 18490 . . 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 17282 . 2 ((𝐹 ∈ Mnd ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
84, 7syl3an1 1357 1 ((𝑊 ∈ SLMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  Scalarcsca 15925  Mndcmnd 17275  SRingcsrg 18486  SLModcslmd 29727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-cmn 18176  df-srg 18487  df-slmd 29728
This theorem is referenced by: (None)
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