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Theorem rngocl 21102
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 21101 . 2  |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
5 fovrn 6032 . 2  |-  ( ( H : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A H B )  e.  X
)
64, 5syl3an1 1215 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 934    = wceq 1633    e. wcel 1701    X. cxp 4724   ran crn 4727   -->wf 5288   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163   RingOpscrngo 21095
This theorem is referenced by:  rngolz  21121  rngorz  21122  rngonegmn1l  25728  rngonegmn1r  25729  rngoneglmul  25730  rngonegrmul  25731  rngosubdi  25732  rngosubdir  25733  isdrngo2  25737  rngohomco  25753  rngoisocnv  25760  crngm4  25776  rngoidl  25797  keridl  25805  prnc  25840  ispridlc  25843  pridlc3  25846  dmncan1  25849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-ov 5903  df-1st 6164  df-2nd 6165  df-rngo 21096
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