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Theorem rngogcl 24031
Description: Closure law for the addition (group) operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringgcl.1  |-  G  =  ( 1st `  R
)
ringgcl.2  |-  X  =  ran  G
Assertion
Ref Expression
rngogcl  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )

Proof of Theorem rngogcl
StepHypRef Expression
1 ringgcl.1 . . 3  |-  G  =  ( 1st `  R
)
21rngogrpo 24030 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringgcl.2 . . 3  |-  X  =  ran  G
43grpocl 23840 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
52, 4syl3an1 1252 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   ran crn 4950   ` cfv 5527  (class class class)co 6201   1stc1st 6686   GrpOpcgr 23826   RingOpscrngo 24015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fo 5533  df-fv 5535  df-ov 6204  df-1st 6688  df-2nd 6689  df-grpo 23831  df-ablo 23922  df-rngo 24016
This theorem is referenced by:  rngohomco  28929  rngoisocnv  28936  rngoidl  28973  keridl  28981  prnc  29016
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