Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  slmdacl Structured version   Visualization version   GIF version

Theorem slmdacl 32937
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 32935 . . 3 (π‘Š ∈ SLMod β†’ 𝐹 ∈ SRing)
3 srgmnd 20137 . . 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 18709 . 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 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  Scalarcsca 17243  Mndcmnd 18701  SRingcsrg 20133  SLModcslmd 32928
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 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-cmn 19744  df-srg 20134  df-slmd 32929
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator