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Theorem rngonegcl 25943
Description: A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
ringnegcl.1  |-  G  =  ( 1st `  R
)
ringnegcl.2  |-  X  =  ran  G
ringnegcl.3  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
rngonegcl  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  e.  X )

Proof of Theorem rngonegcl
StepHypRef Expression
1 ringnegcl.1 . . 3  |-  G  =  ( 1st `  R
)
21rngogrpo 21017 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringnegcl.2 . . 3  |-  X  =  ran  G
4 ringnegcl.3 . . 3  |-  N  =  ( inv `  G
)
53, 4grpoinvcl 20853 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
62, 5sylan 459 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   ran crn 4662   ` cfv 4673   1stc1st 6054   GrpOpcgr 20813   invcgn 20815   RingOpscrngo 21002
This theorem is referenced by:  rngonegmn1l  25947  rngonegmn1r  25948  rngoneglmul  25949  rngonegrmul  25950  rngosubdi  25951  rngosubdir  25952  idlnegcl  26014
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-grpo 20818  df-gid 20819  df-ginv 20820  df-ablo 20909  df-rngo 21003
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