Theorem srgacl 18524

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 e. SRing /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )

Proof of Theorem srgacl

Step	Hyp	Ref	Expression
1		srgmnd 18509	. 2 |- ( R e. SRing -> R e. Mnd )
2		srgacl.b	. . 3 |- B = ( Base ` R )
3		srgacl.p	. . 3 |- .+ = ( +g ` R )
4	2, 3	mndcl 17301	. 2 |- ( ( R e. Mnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
5	1, 4	syl3an1 1359	1 |- ( ( R e. SRing /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294  SRingcsrg 18505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cmn 18195  df-srg 18506

This theorem is referenced by:  srglmhm  18535  srgrmhm  18536  sge0tsms  40597