MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  srgacl Structured version   Unicode version

Theorem srgacl 17493
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
StepHypRef Expression
1 srgmnd 17479 . 2  |-  ( R  e. SRing  ->  R  e.  Mnd )
2 srgacl.b . . 3  |-  B  =  ( Base `  R
)
3 srgacl.p . . 3  |-  .+  =  ( +g  `  R )
42, 3mndcl 16251 . 2  |-  ( ( R  e.  Mnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y
)  e.  B )
51, 4syl3an1 1263 1  |-  ( ( R  e. SRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277   Basecbs 14839   +g cplusg 14907   Mndcmnd 16241  SRingcsrg 17475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-nul 4524
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5532  df-fv 5576  df-ov 6280  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-cmn 17122  df-srg 17476
This theorem is referenced by:  srglmhm  17504  srgrmhm  17505
  Copyright terms: Public domain W3C validator