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Theorem rngocl 25362
Description: Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1  |-  G  =  ( 1st `  R
)
ringi.2  |-  H  =  ( 2nd `  R
)
ringi.3  |-  X  =  ran  G
Assertion
Ref Expression
rngocl  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )

Proof of Theorem rngocl
StepHypRef Expression
1 ringi.1 . . 3  |-  G  =  ( 1st `  R
)
2 ringi.2 . . 3  |-  H  =  ( 2nd `  R
)
3 ringi.3 . . 3  |-  X  =  ran  G
41, 2, 3rngosm 25361 . 2  |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
5 fovrn 6430 . 2  |-  ( ( H : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A H B )  e.  X
)
64, 5syl3an1 1262 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1383    e. wcel 1804    X. cxp 4987   ran crn 4990   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   RingOpscrngo 25355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fv 5586  df-ov 6284  df-1st 6785  df-2nd 6786  df-rngo 25356
This theorem is referenced by:  rngolz  25381  rngorz  25382  rngonegmn1l  30328  rngonegmn1r  30329  rngoneglmul  30330  rngonegrmul  30331  rngosubdi  30332  rngosubdir  30333  isdrngo2  30337  rngohomco  30353  rngoisocnv  30360  crngm4  30376  rngoidl  30397  keridl  30405  prnc  30440  ispridlc  30443  pridlc3  30446  dmncan1  30449
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