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Theorem unitnegcl 15706
Description: The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
unitnegcl.1  |-  U  =  (Unit `  R )
unitnegcl.2  |-  N  =  ( inv g `  R )
Assertion
Ref Expression
unitnegcl  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )  e.  U )

Proof of Theorem unitnegcl
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  R  e.  Ring )
2 rnggrp 15589 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
3 eqid 2380 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
4 unitnegcl.1 . . . . . . 7  |-  U  =  (Unit `  R )
53, 4unitcl 15684 . . . . . 6  |-  ( X  e.  U  ->  X  e.  ( Base `  R
) )
6 unitnegcl.2 . . . . . . 7  |-  N  =  ( inv g `  R )
73, 6grpinvcl 14770 . . . . . 6  |-  ( ( R  e.  Grp  /\  X  e.  ( Base `  R ) )  -> 
( N `  X
)  e.  ( Base `  R ) )
82, 5, 7syl2an 464 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )  e.  ( Base `  R
) )
9 eqid 2380 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
103, 9, 6dvdsrneg 15679 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N `  X )  e.  ( Base `  R
) )  ->  ( N `  X )
( ||r `
 R ) ( N `  ( N `
 X ) ) )
118, 10syldan 457 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 R ) ( N `  ( N `
 X ) ) )
123, 6grpinvinv 14778 . . . . 5  |-  ( ( R  e.  Grp  /\  X  e.  ( Base `  R ) )  -> 
( N `  ( N `  X )
)  =  X )
132, 5, 12syl2an 464 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  ( N `  X ) )  =  X )
1411, 13breqtrd 4170 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 R ) X )
15 simpr 448 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  X  e.  U )
16 eqid 2380 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
17 eqid 2380 . . . . . 6  |-  (oppr `  R
)  =  (oppr `  R
)
18 eqid 2380 . . . . . 6  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
194, 16, 9, 17, 18isunit 15682 . . . . 5  |-  ( X  e.  U  <->  ( X
( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2015, 19sylib 189 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( X ( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2120simpld 446 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  X
( ||r `
 R ) ( 1r `  R ) )
223, 9dvdsrtr 15677 . . 3  |-  ( ( R  e.  Ring  /\  ( N `  X )
( ||r `
 R ) X  /\  X ( ||r `  R
) ( 1r `  R ) )  -> 
( N `  X
) ( ||r `
 R ) ( 1r `  R ) )
231, 14, 21, 22syl3anc 1184 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 R ) ( 1r `  R ) )
2417opprrng 15656 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
2524adantr 452 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (oppr `  R
)  e.  Ring )
2617, 3opprbas 15654 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
2717, 6opprneg 15660 . . . . . 6  |-  N  =  ( inv g `  (oppr `  R ) )
2826, 18, 27dvdsrneg 15679 . . . . 5  |-  ( ( (oppr
`  R )  e. 
Ring  /\  ( N `  X )  e.  (
Base `  R )
)  ->  ( N `  X ) ( ||r `  (oppr `  R
) ) ( N `
 ( N `  X ) ) )
2925, 8, 28syl2anc 643 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) ( N `  ( N `  X )
) )
3029, 13breqtrd 4170 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) X )
3120simprd 450 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  X
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
3226, 18dvdsrtr 15677 . . 3  |-  ( ( (oppr
`  R )  e. 
Ring  /\  ( N `  X ) ( ||r `  (oppr `  R
) ) X  /\  X ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
3325, 30, 31, 32syl3anc 1184 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
344, 16, 9, 17, 18isunit 15682 . 2  |-  ( ( N `  X )  e.  U  <->  ( ( N `  X )
( ||r `
 R ) ( 1r `  R )  /\  ( N `  X ) ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
3523, 33, 34sylanbrc 646 1  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4146   ` cfv 5387   Basecbs 13389   Grpcgrp 14605   inv gcminusg 14606   Ringcrg 15580   1rcur 15582  opprcoppr 15647   ||rcdsr 15663  Unitcui 15664
This theorem is referenced by:  irredneg  15735  deg1invg  19889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-tpos 6408  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-plusg 13462  df-mulr 13463  df-0g 13647  df-mnd 14610  df-grp 14732  df-minusg 14733  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667
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