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Theorem rngocl 25585
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 25584 . 2  |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
5 fovrn 6418 . 2  |-  ( ( H : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A H B )  e.  X
)
64, 5syl3an1 1259 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 971    = wceq 1398    e. wcel 1823    X. cxp 4986   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   RingOpscrngo 25578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-1st 6773  df-2nd 6774  df-rngo 25579
This theorem is referenced by:  rngolz  25604  rngorz  25605  rngonegmn1l  30595  rngonegmn1r  30596  rngoneglmul  30597  rngonegrmul  30598  rngosubdi  30599  rngosubdir  30600  isdrngo2  30604  rngohomco  30620  rngoisocnv  30627  crngm4  30643  rngoidl  30664  keridl  30672  prnc  30707  ispridlc  30710  pridlc3  30713  dmncan1  30716
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