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Theorem srgacl 18732
 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 18717 . 2 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
2 srgacl.b . . 3 𝐵 = (Base‘𝑅)
3 srgacl.p . . 3 + = (+g𝑅)
42, 3mndcl 17509 . 2 ((𝑅 ∈ Mnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
51, 4syl3an1 1166 1 ((𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  Mndcmnd 17502  SRingcsrg 18713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-cmn 18402  df-srg 18714 This theorem is referenced by:  srglmhm  18743  srgrmhm  18744  sge0tsms  41114
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