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Theorem srgacl 19895
Description: Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
srgacl.b 𝐵 = (Base‘𝑅)
srgacl.p + = (+g𝑅)
Assertion
Ref Expression
srgacl ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Proof of Theorem srgacl
StepHypRef Expression
1 srgmnd 19880 . 2 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
2 srgacl.b . . 3 𝐵 = (Base‘𝑅)
3 srgacl.p . . 3 + = (+g𝑅)
42, 3mndcl 18524 . 2 ((𝑅 ∈ Mnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
51, 4syl3an1 1163 1 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cfv 6493  (class class class)co 7351  Basecbs 17043  +gcplusg 17093  Mndcmnd 18516  SRingcsrg 19876
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-nul 5261
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-ov 7354  df-mgm 18457  df-sgrp 18506  df-mnd 18517  df-cmn 19523  df-srg 19877
This theorem is referenced by:  srglmhm  19906  srgrmhm  19907  sge0tsms  44522
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