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Theorem slmdacl 32857
Description: Closure of ring addition for a semimodule. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmdacl.f 𝐹 = (Scalarβ€˜π‘Š)
slmdacl.k 𝐾 = (Baseβ€˜πΉ)
slmdacl.p + = (+gβ€˜πΉ)
Assertion
Ref Expression
slmdacl ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝐾) β†’ (𝑋 + π‘Œ) ∈ 𝐾)

Proof of Theorem slmdacl
StepHypRef Expression
1 slmdacl.f . . . 4 𝐹 = (Scalarβ€˜π‘Š)
21slmdsrg 32855 . . 3 (π‘Š ∈ SLMod β†’ 𝐹 ∈ SRing)
3 srgmnd 20092 . . 3 (𝐹 ∈ SRing β†’ 𝐹 ∈ Mnd)
42, 3syl 17 . 2 (π‘Š ∈ SLMod β†’ 𝐹 ∈ Mnd)
5 slmdacl.k . . 3 𝐾 = (Baseβ€˜πΉ)
6 slmdacl.p . . 3 + = (+gβ€˜πΉ)
75, 6mndcl 18672 . 2 ((𝐹 ∈ Mnd ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝐾) β†’ (𝑋 + π‘Œ) ∈ 𝐾)
84, 7syl3an1 1160 1 ((π‘Š ∈ SLMod ∧ 𝑋 ∈ 𝐾 ∧ π‘Œ ∈ 𝐾) β†’ (𝑋 + π‘Œ) ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  +gcplusg 17203  Scalarcsca 17206  Mndcmnd 18664  SRingcsrg 20088  SLModcslmd 32848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-cmn 19699  df-srg 20089  df-slmd 32849
This theorem is referenced by: (None)
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