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Theorem rngacl 15384
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 15362 . 2  |-  ( R  e.  Ring  ->  R  e. 
Grp )
2 rngacl.b . . 3  |-  B  =  ( Base `  R
)
3 rngacl.p . . 3  |-  .+  =  ( +g  `  R )
42, 3grpcl 14511 . 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   Ringcrg 15353
This theorem is referenced by:  rngcom  15385  rnglghm  15404  rngrghm  15405  imasrng  15418  divsrng2  15419  cntzsubr  15593  srngadd  15638  issrngd  15642  lmodprop2d  15703  prdslmodd  15742  psrlmod  16162  coe1add  16357  ip2subdi  16564  mpfind  19444  mdegaddle  19476  deg1addle2  19504  deg1add  19505  ply1divex  19538  mendlmod  27604  dvhlveclem  31920  baerlem3lem1  32519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165  ax-pow 4204
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-mnd 14383  df-grp 14505  df-rng 15356
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