Theorem unitnegcl 14275

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

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( R e. Ring /\ X e. U ) -> R e. Ring )
2		ringgrp 14145	. . . . . 6 |- ( R e. Ring -> R e. Grp )
3		eqidd 2233	. . . . . . 7 |- ( ( R e. Ring /\ X e. U ) -> ( Base ` R ) = ( Base ` R ) )
4		unitnegcl.1	. . . . . . . 8 |- U = (Unit ` R )
5	4	a1i 9	. . . . . . 7 |- ( ( R e. Ring /\ X e. U ) -> U = (Unit ` R ) )
6		ringsrg 14191	. . . . . . . 8 |- ( R e. Ring -> R e. SRing )
7	6	adantr 276	. . . . . . 7 |- ( ( R e. Ring /\ X e. U ) -> R e. SRing )
8		simpr 110	. . . . . . 7 |- ( ( R e. Ring /\ X e. U ) -> X e. U )
9	3, 5, 7, 8	unitcld 14253	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> X e. ( Base ` R ) )
10		eqid 2232	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
11		unitnegcl.2	. . . . . . 7 |- N = ( invg ` R )
12	10, 11	grpinvcl 13761	. . . . . 6 |- ( ( R e. Grp /\ X e. ( Base ` R ) ) -> ( N ` X ) e. ( Base ` R ) )
13	2, 9, 12	syl2an2r 599	. . . . 5 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) e. ( Base ` R ) )
14		eqid 2232	. . . . . 6 |- ( ||r ` R ) = ( ||r ` R )
15	10, 14, 11	dvdsrneg 14248	. . . . 5 |- ( ( R e. Ring /\ ( N ` X ) e. ( Base ` R ) ) -> ( N ` X ) ( ||r ` R ) ( N ` ( N ` X ) ) )
16	13, 15	syldan 282	. . . 4 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` R ) ( N ` ( N ` X ) ) )
17	10, 11	grpinvinv 13780	. . . . 5 |- ( ( R e. Grp /\ X e. ( Base ` R ) ) -> ( N ` ( N ` X ) ) = X )
18	2, 9, 17	syl2an2r 599	. . . 4 |- ( ( R e. Ring /\ X e. U ) -> ( N ` ( N ` X ) ) = X )
19	16, 18	breqtrd 4135	. . 3 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` R ) X )
20		eqidd 2233	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
21		eqidd 2233	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
22		eqidd 2233	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> (oppr ` R ) = (oppr ` R ) )
23		eqidd 2233	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
24	5, 20, 21, 22, 23, 7	isunitd 14251	. . . . 5 |- ( ( R e. Ring /\ X e. U ) -> ( X e. U <-> ( X ( ||r ` R ) ( 1r ` R ) /\ X ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
25	8, 24	mpbid 147	. . . 4 |- ( ( R e. Ring /\ X e. U ) -> ( X ( ||r ` R ) ( 1r ` R ) /\ X ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) )
26	25	simpld 112	. . 3 |- ( ( R e. Ring /\ X e. U ) -> X ( ||r ` R ) ( 1r ` R ) )
27	10, 14	dvdsrtr 14246	. . 3 |- ( ( R e. Ring /\ ( N ` X ) ( ||r ` R ) X /\ X ( ||r ` R ) ( 1r ` R ) ) -> ( N ` X ) ( ||r ` R ) ( 1r ` R ) )
28	1, 19, 26, 27	syl3anc 1274	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` R ) ( 1r ` R ) )
29		eqid 2232	. . . . 5 |- (oppr ` R ) = (oppr ` R )
30	29	opprring 14223	. . . 4 |- ( R e. Ring -> (oppr ` R ) e. Ring )
31	30	adantr 276	. . 3 |- ( ( R e. Ring /\ X e. U ) -> (oppr ` R ) e. Ring )
32	29, 10	opprbasg 14219	. . . . . . . . 9 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
33	32	eleq2d 2302	. . . . . . . 8 |- ( R e. Ring -> ( ( N ` X ) e. ( Base ` R ) <-> ( N ` X ) e. ( Base ` (oppr ` R ) ) ) )
34	33	adantr 276	. . . . . . 7 |- ( ( R e. Ring /\ X e. U ) -> ( ( N ` X ) e. ( Base ` R ) <-> ( N ` X ) e. ( Base ` (oppr ` R ) ) ) )
35	13, 34	mpbid 147	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) e. ( Base ` (oppr ` R ) ) )
36		eqid 2232	. . . . . . 7 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
37		eqid 2232	. . . . . . 7 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
38		eqid 2232	. . . . . . 7 |- ( invg ` (oppr ` R ) ) = ( invg ` (oppr ` R ) )
39	36, 37, 38	dvdsrneg 14248	. . . . . 6 |- ( ( (oppr ` R ) e. Ring /\ ( N ` X ) e. ( Base ` (oppr ` R ) ) ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) ( ( invg ` (oppr ` R ) ) ` ( N ` X ) ) )
40	30, 35, 39	syl2an2r 599	. . . . 5 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) ( ( invg ` (oppr ` R ) ) ` ( N ` X ) ) )
41	29, 11	opprnegg 14227	. . . . . . . 8 |- ( R e. Ring -> N = ( invg ` (oppr ` R ) ) )
42	41	fveq1d 5672	. . . . . . 7 |- ( R e. Ring -> ( N ` ( N ` X ) ) = ( ( invg ` (oppr ` R ) ) ` ( N ` X ) ) )
43	42	breq2d 4121	. . . . . 6 |- ( R e. Ring -> ( ( N ` X ) ( ||r ` (oppr ` R ) ) ( N ` ( N ` X ) ) <-> ( N ` X ) ( ||r ` (oppr ` R ) ) ( ( invg ` (oppr ` R ) ) ` ( N ` X ) ) ) )
44	43	adantr 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 ) ) ) )
45	40, 44	mpbird 167	. . . 4 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) ( N ` ( N ` X ) ) )
46	45, 18	breqtrd 4135	. . 3 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) X )
47	25	simprd 114	. . 3 |- ( ( R e. Ring /\ X e. U ) -> X ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
48	36, 37	dvdsrtr 14246	. . 3 |- ( ( (oppr ` R ) e. Ring /\ ( N ` X ) ( ||r ` (oppr ` R ) ) X /\ X ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
49	31, 46, 47, 48	syl3anc 1274	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
50	5, 20, 21, 22, 23, 7	isunitd 14251	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( ( N ` X ) e. U <-> ( ( N ` X ) ( ||r ` R ) ( 1r ` R ) /\ ( N ` X ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) ) ) )
51	28, 49, 50	mpbir2and 953	1 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) e. U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   class class class wbr 4109   ` cfv 5352   Basecbs 13212   Grpcgrp 13713   invgcminusg 13714   1rcur 14103  SRingcsrg 14107   Ringcrg 14140  opp_rcoppr 14211   ||_rcdsr 14230  Unitcui 14231

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-tpos 6476  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-minusg 13717  df-cmn 14003  df-abl 14004  df-mgp 14065  df-ur 14104  df-srg 14108  df-ring 14142  df-oppr 14212  df-dvdsr 14233  df-unit 14234

This theorem is referenced by:  aprsym  14430