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Theorem rngonegcl 25742
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 20887 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringnegcl.2 . . 3  |-  X  =  ran  G
4 ringnegcl.3 . . 3  |-  N  =  ( inv `  G
)
53, 4grpoinvcl 20723 . 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 4581   ` cfv 4592   1stc1st 5972   GrpOpcgr 20683   invcgn 20685   RingOpscrngo 20872
This theorem is referenced by:  rngonegmn1l  25746  rngonegmn1r  25747  rngoneglmul  25748  rngonegrmul  25749  rngosubdi  25750  rngosubdir  25751  idlnegcl  25813
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-grpo 20688  df-gid 20689  df-ginv 20690  df-ablo 20779  df-rngo 20873
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