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Theorem rngocl 24014
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 24013 . 2  |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
5 fovrn 6336 . 2  |-  ( ( H : ( X  X.  X ) --> X  /\  A  e.  X  /\  B  e.  X
)  ->  ( A H B )  e.  X
)
64, 5syl3an1 1252 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 965    = wceq 1370    e. wcel 1758    X. cxp 4939   ran crn 4942   -->wf 5515   ` cfv 5519  (class class class)co 6193   1stc1st 6678   2ndc2nd 6679   RingOpscrngo 24007
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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-1st 6680  df-2nd 6681  df-rngo 24008
This theorem is referenced by:  rngolz  24033  rngorz  24034  rngonegmn1l  28896  rngonegmn1r  28897  rngoneglmul  28898  rngonegrmul  28899  rngosubdi  28900  rngosubdir  28901  isdrngo2  28905  rngohomco  28921  rngoisocnv  28928  crngm4  28944  rngoidl  28965  keridl  28973  prnc  29008  ispridlc  29011  pridlc3  29014  dmncan1  29017
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