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Theorem slmdacl 30868
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 30866 . . 3 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 srgmnd 19250 . . 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 17910 . 2 ((𝐹 ∈ Mnd ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
84, 7syl3an1 1160 1 ((𝑊 ∈ SLMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2114  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  Scalarcsca 16559  Mndcmnd 17902  SRingcsrg 19246  SLModcslmd 30859
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-cmn 18899  df-srg 19247  df-slmd 30860
This theorem is referenced by: (None)
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