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Theorem rngacl 15368
Description: Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.)
Hypotheses
Ref Expression
rngacl.b  |-  B  =  ( Base `  R
)
rngacl.p  |-  .+  =  ( +g  `  R )
Assertion
Ref Expression
rngacl  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )

Proof of Theorem rngacl
StepHypRef Expression
1 rnggrp 15346 . 2  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2 rngacl.b . . 3  |-  B  =  ( Base `  R
)
3 rngacl.p . . 3  |-  .+  =  ( +g  `  R )
42, 3grpcl 14495 . 2  |-  ( ( R  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y
)  e.  B )
51, 4syl3an1 1215 1  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   Ringcrg 15337
This theorem is referenced by:  rngcom  15369  rnglghm  15388  rngrghm  15389  imasrng  15402  divsrng2  15403  cntzsubr  15577  srngadd  15622  issrngd  15626  lmodprop2d  15687  prdslmodd  15726  psrlmod  16146  coe1add  16341  ip2subdi  16548  mpfind  19428  mdegaddle  19460  deg1addle2  19488  deg1add  19489  ply1divex  19522  mendlmod  27501  dvhlveclem  31298  baerlem3lem1  31897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149  ax-pow 4188
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-mnd 14367  df-grp 14489  df-rng 15340
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