MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unitnegcl Structured version   Unicode version

Theorem unitnegcl 17650
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  =  ( invg `  R )
Assertion
Ref Expression
unitnegcl  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )  e.  U )

Proof of Theorem unitnegcl
StepHypRef Expression
1 simpl 455 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  R  e.  Ring )
2 ringgrp 17523 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
3 eqid 2402 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
4 unitnegcl.1 . . . . . . 7  |-  U  =  (Unit `  R )
53, 4unitcl 17628 . . . . . 6  |-  ( X  e.  U  ->  X  e.  ( Base `  R
) )
6 unitnegcl.2 . . . . . . 7  |-  N  =  ( invg `  R )
73, 6grpinvcl 16419 . . . . . 6  |-  ( ( R  e.  Grp  /\  X  e.  ( Base `  R ) )  -> 
( N `  X
)  e.  ( Base `  R ) )
82, 5, 7syl2an 475 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )  e.  ( Base `  R
) )
9 eqid 2402 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
103, 9, 6dvdsrneg 17623 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N `  X )  e.  ( Base `  R
) )  ->  ( N `  X )
( ||r `
 R ) ( N `  ( N `
 X ) ) )
118, 10syldan 468 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 R ) ( N `  ( N `
 X ) ) )
123, 6grpinvinv 16429 . . . . 5  |-  ( ( R  e.  Grp  /\  X  e.  ( Base `  R ) )  -> 
( N `  ( N `  X )
)  =  X )
132, 5, 12syl2an 475 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  ( N `  X ) )  =  X )
1411, 13breqtrd 4419 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 R ) X )
15 simpr 459 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  X  e.  U )
16 eqid 2402 . . . . . 6  |-  ( 1r
`  R )  =  ( 1r `  R
)
17 eqid 2402 . . . . . 6  |-  (oppr `  R
)  =  (oppr `  R
)
18 eqid 2402 . . . . . 6  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
194, 16, 9, 17, 18isunit 17626 . . . . 5  |-  ( X  e.  U  <->  ( X
( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2015, 19sylib 196 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( X ( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2120simpld 457 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  X
( ||r `
 R ) ( 1r `  R ) )
223, 9dvdsrtr 17621 . . 3  |-  ( ( R  e.  Ring  /\  ( N `  X )
( ||r `
 R ) X  /\  X ( ||r `  R
) ( 1r `  R ) )  -> 
( N `  X
) ( ||r `
 R ) ( 1r `  R ) )
231, 14, 21, 22syl3anc 1230 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 R ) ( 1r `  R ) )
2417opprring 17600 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
2524adantr 463 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (oppr `  R
)  e.  Ring )
2617, 3opprbas 17598 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
2717, 6opprneg 17604 . . . . . 6  |-  N  =  ( invg `  (oppr `  R ) )
2826, 18, 27dvdsrneg 17623 . . . . 5  |-  ( ( (oppr
`  R )  e. 
Ring  /\  ( N `  X )  e.  (
Base `  R )
)  ->  ( N `  X ) ( ||r `  (oppr `  R
) ) ( N `
 ( N `  X ) ) )
2925, 8, 28syl2anc 659 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) ( N `  ( N `  X )
) )
3029, 13breqtrd 4419 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) X )
3120simprd 461 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  X
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
3226, 18dvdsrtr 17621 . . 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 1230 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
344, 16, 9, 17, 18isunit 17626 . 2  |-  ( ( N `  X )  e.  U  <->  ( ( N `  X )
( ||r `
 R ) ( 1r `  R )  /\  ( N `  X ) ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
3523, 33, 34sylanbrc 662 1  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   class class class wbr 4395   ` cfv 5569   Basecbs 14841   Grpcgrp 16377   invgcminusg 16378   1rcur 17473   Ringcrg 17518  opprcoppr 17591   ||rcdsr 17607  Unitcui 17608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-plusg 14922  df-mulr 14923  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-minusg 16382  df-mgp 17462  df-ur 17474  df-ring 17520  df-oppr 17592  df-dvdsr 17610  df-unit 17611
This theorem is referenced by:  irredneg  17679  deg1invg  22799  nzrneg1ne0  38186  invginvrid  38471  lincresunit3lem3  38586  lincresunitlem1  38587
  Copyright terms: Public domain W3C validator