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Theorem unitnegcl 15778
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 15661 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
3 eqid 2435 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
4 unitnegcl.1 . . . . . . 7  |-  U  =  (Unit `  R )
53, 4unitcl 15756 . . . . . 6  |-  ( X  e.  U  ->  X  e.  ( Base `  R
) )
6 unitnegcl.2 . . . . . . 7  |-  N  =  ( inv g `  R )
73, 6grpinvcl 14842 . . . . . 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 2435 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
103, 9, 6dvdsrneg 15751 . . . . 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 14850 . . . . 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 4228 . . 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 2435 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
17 eqid 2435 . . . . . 6  |-  (oppr `  R
)  =  (oppr `  R
)
18 eqid 2435 . . . . . 6  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
194, 16, 9, 17, 18isunit 15754 . . . . 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 15749 . . 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 15728 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
2524adantr 452 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (oppr `  R
)  e.  Ring )
2617, 3opprbas 15726 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
2717, 6opprneg 15732 . . . . . 6  |-  N  =  ( inv g `  (oppr `  R ) )
2826, 18, 27dvdsrneg 15751 . . . . 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 4228 . . 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 15749 . . 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 15754 . 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 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13461   Grpcgrp 14677   inv gcminusg 14678   Ringcrg 15652   1rcur 15654  opprcoppr 15719   ||rcdsr 15735  Unitcui 15736
This theorem is referenced by:  irredneg  15807  deg1invg  20021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-mulr 13535  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739
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