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Theorem unitnegcl 13299
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 109 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  R  e.  Ring )
2 ringgrp 13184 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
3 eqidd 2178 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( Base `  R )  =  ( Base `  R
) )
4 unitnegcl.1 . . . . . . . 8  |-  U  =  (Unit `  R )
54a1i 9 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  U  =  (Unit `  R )
)
6 ringsrg 13224 . . . . . . . 8  |-  ( R  e.  Ring  ->  R  e. SRing
)
76adantr 276 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  R  e. SRing )
8 simpr 110 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  X  e.  U )
93, 5, 7, 8unitcld 13277 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  X  e.  ( Base `  R
) )
10 eqid 2177 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
11 unitnegcl.2 . . . . . . 7  |-  N  =  ( invg `  R )
1210, 11grpinvcl 12921 . . . . . 6  |-  ( ( R  e.  Grp  /\  X  e.  ( Base `  R ) )  -> 
( N `  X
)  e.  ( Base `  R ) )
132, 9, 12syl2an2r 595 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )  e.  ( Base `  R
) )
14 eqid 2177 . . . . . 6  |-  ( ||r `  R
)  =  ( ||r `  R
)
1510, 14, 11dvdsrneg 13272 . . . . 5  |-  ( ( R  e.  Ring  /\  ( N `  X )  e.  ( Base `  R
) )  ->  ( N `  X )
( ||r `
 R ) ( N `  ( N `
 X ) ) )
1613, 15syldan 282 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 R ) ( N `  ( N `
 X ) ) )
1710, 11grpinvinv 12937 . . . . 5  |-  ( ( R  e.  Grp  /\  X  e.  ( Base `  R ) )  -> 
( N `  ( N `  X )
)  =  X )
182, 9, 17syl2an2r 595 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  ( N `  X ) )  =  X )
1916, 18breqtrd 4030 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 R ) X )
20 eqidd 2178 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( 1r `  R )  =  ( 1r `  R
) )
21 eqidd 2178 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( ||r `  R )  =  (
||r `  R ) )
22 eqidd 2178 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (oppr `  R
)  =  (oppr `  R
) )
23 eqidd 2178 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( ||r `  (oppr
`  R ) )  =  ( ||r `
 (oppr
`  R ) ) )
245, 20, 21, 22, 23, 7isunitd 13275 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( X  e.  U  <->  ( X
( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) ) )
258, 24mpbid 147 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( X ( ||r `
 R ) ( 1r `  R )  /\  X ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) )
2625simpld 112 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  X
( ||r `
 R ) ( 1r `  R ) )
2710, 14dvdsrtr 13270 . . 3  |-  ( ( R  e.  Ring  /\  ( N `  X )
( ||r `
 R ) X  /\  X ( ||r `  R
) ( 1r `  R ) )  -> 
( N `  X
) ( ||r `
 R ) ( 1r `  R ) )
281, 19, 26, 27syl3anc 1238 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 R ) ( 1r `  R ) )
29 eqid 2177 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
3029opprring 13249 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
3130adantr 276 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (oppr `  R
)  e.  Ring )
3229, 10opprbasg 13247 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  (oppr
`  R ) ) )
3332eleq2d 2247 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( ( N `  X )  e.  ( Base `  R
)  <->  ( N `  X )  e.  (
Base `  (oppr
`  R ) ) ) )
3433adantr 276 . . . . . . 7  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (
( N `  X
)  e.  ( Base `  R )  <->  ( N `  X )  e.  (
Base `  (oppr
`  R ) ) ) )
3513, 34mpbid 147 . . . . . 6  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )  e.  ( Base `  (oppr `  R
) ) )
36 eqid 2177 . . . . . . 7  |-  ( Base `  (oppr
`  R ) )  =  ( Base `  (oppr `  R
) )
37 eqid 2177 . . . . . . 7  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
38 eqid 2177 . . . . . . 7  |-  ( invg `  (oppr `  R
) )  =  ( invg `  (oppr `  R
) )
3936, 37, 38dvdsrneg 13272 . . . . . 6  |-  ( ( (oppr
`  R )  e. 
Ring  /\  ( N `  X )  e.  (
Base `  (oppr
`  R ) ) )  ->  ( N `  X ) ( ||r `  (oppr `  R
) ) ( ( invg `  (oppr `  R
) ) `  ( N `  X )
) )
4030, 35, 39syl2an2r 595 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) ( ( invg `  (oppr
`  R ) ) `
 ( N `  X ) ) )
4129, 11opprnegg 13253 . . . . . . . 8  |-  ( R  e.  Ring  ->  N  =  ( invg `  (oppr `  R ) ) )
4241fveq1d 5518 . . . . . . 7  |-  ( R  e.  Ring  ->  ( N `
 ( N `  X ) )  =  ( ( invg `  (oppr
`  R ) ) `
 ( N `  X ) ) )
4342breq2d 4016 . . . . . 6  |-  ( R  e.  Ring  ->  ( ( N `  X ) ( ||r `
 (oppr
`  R ) ) ( N `  ( N `  X )
)  <->  ( N `  X ) ( ||r `  (oppr `  R
) ) ( ( invg `  (oppr `  R
) ) `  ( N `  X )
) ) )
4443adantr 276 . . . . 5  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (
( N `  X
) ( ||r `
 (oppr
`  R ) ) ( N `  ( N `  X )
)  <->  ( N `  X ) ( ||r `  (oppr `  R
) ) ( ( invg `  (oppr `  R
) ) `  ( N `  X )
) ) )
4540, 44mpbird 167 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) ( N `  ( N `  X )
) )
4645, 18breqtrd 4030 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) X )
4725simprd 114 . . 3  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  X
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
4836, 37dvdsrtr 13270 . . 3  |-  ( ( (oppr
`  R )  e. 
Ring  /\  ( N `  X ) ( ||r `  (oppr `  R
) ) X  /\  X ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
4931, 46, 47, 48syl3anc 1238 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )
( ||r `
 (oppr
`  R ) ) ( 1r `  R
) )
505, 20, 21, 22, 23, 7isunitd 13275 . 2  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  (
( N `  X
)  e.  U  <->  ( ( N `  X )
( ||r `
 R ) ( 1r `  R )  /\  ( N `  X ) ( ||r `  (oppr `  R
) ) ( 1r
`  R ) ) ) )
5128, 49, 50mpbir2and 944 1  |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( N `  X )  e.  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   class class class wbr 4004   ` cfv 5217   Basecbs 12462   Grpcgrp 12877   invgcminusg 12878   1rcur 13142  SRingcsrg 13146   Ringcrg 13179  opprcoppr 13239   ||rcdsr 13255  Unitcui 13256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-lttrn 7925  ax-pre-ltadd 7927
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-tpos 6246  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-plusg 12549  df-mulr 12550  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-cmn 13090  df-abl 13091  df-mgp 13131  df-ur 13143  df-srg 13147  df-ring 13181  df-oppr 13240  df-dvdsr 13258  df-unit 13259
This theorem is referenced by:  aprsym  13374
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