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Theorem rngocl 21975
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 21974 . 2  |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
5 fovrn 6219 . 2  |-  ( ( H : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A H B )  e.  X
)
64, 5syl3an1 1218 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    X. cxp 4879   ran crn 4882   -->wf 5453   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   RingOpscrngo 21968
This theorem is referenced by:  rngolz  21994  rngorz  21995  rngonegmn1l  26579  rngonegmn1r  26580  rngoneglmul  26581  rngonegrmul  26582  rngosubdi  26583  rngosubdir  26584  isdrngo2  26588  rngohomco  26604  rngoisocnv  26611  crngm4  26627  rngoidl  26648  keridl  26656  prnc  26691  ispridlc  26694  pridlc3  26697  dmncan1  26700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353  df-rngo 21969
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