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Theorem rngonegcl 31594
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 25687 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringnegcl.2 . . 3  |-  X  =  ran  G
4 ringnegcl.3 . . 3  |-  N  =  ( inv `  G
)
53, 4grpoinvcl 25523 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
62, 5sylan 469 1  |-  ( ( R  e.  RingOps  /\  A  e.  X )  ->  ( N `  A )  e.  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1403    e. wcel 1840   ran crn 4941   ` cfv 5523   1stc1st 6734   GrpOpcgr 25483   invcgn 25485   RingOpscrngo 25672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-1st 6736  df-2nd 6737  df-grpo 25488  df-gid 25489  df-ginv 25490  df-ablo 25579  df-rngo 25673
This theorem is referenced by:  rngonegmn1l  31598  rngonegmn1r  31599  rngoneglmul  31600  rngonegrmul  31601  rngosubdi  31602  rngosubdir  31603  idlnegcl  31665
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