Theorem srgacl 20114

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	⊢ + = (+g‘𝑅)

Assertion

Ref	Expression
srgacl	⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Proof of Theorem srgacl

Step	Hyp	Ref	Expression
1		srgmnd 20099	. 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
2		srgacl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		srgacl.p	. . 3 ⊢ + = (+g‘𝑅)
4	2, 3	mndcl 18669	. 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
5	1, 4	syl3an1 1163	1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  Mndcmnd 18661  SRingcsrg 20095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-cmn 19712  df-srg 20096

This theorem is referenced by:  srgcom4lem  20122  srgcom4  20123  srglmhm  20130  srgrmhm  20131  sge0tsms  46378