Theorem unitnegcl 14486

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 14354	. . . . . 6 |- ( R e. Ring -> R e. Grp )
3		eqidd 2239	. . . . . . 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 14401	. . . . . . . 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 14464	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> X e. ( Base ` R ) )
10		eqid 2238	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
11		unitnegcl.2	. . . . . . 7 |- N = ( invg ` R )
12	10, 11	grpinvcl 13902	. . . . . 6 |- ( ( R e. Grp /\ X e. ( Base ` R ) ) -> ( N ` X ) e. ( Base ` R ) )
13	2, 9, 12	syl2an2r 603	. . . . 5 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) e. ( Base ` R ) )
14		eqid 2238	. . . . . 6 |- ( ||r ` R ) = ( ||r ` R )
15	10, 14, 11	dvdsrneg 14459	. . . . 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 13921	. . . . 5 |- ( ( R e. Grp /\ X e. ( Base ` R ) ) -> ( N ` ( N ` X ) ) = X )
18	2, 9, 17	syl2an2r 603	. . . 4 |- ( ( R e. Ring /\ X e. U ) -> ( N ` ( N ` X ) ) = X )
19	16, 18	breqtrd 4156	. . 3 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` R ) X )
20		eqidd 2239	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
21		eqidd 2239	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
22		eqidd 2239	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> (oppr ` R ) = (oppr ` R ) )
23		eqidd 2239	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
24	5, 20, 21, 22, 23, 7	isunitd 14462	. . . . 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 14457	. . 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 1278	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` R ) ( 1r ` R ) )
29		eqid 2238	. . . . 5 |- (oppr ` R ) = (oppr ` R )
30	29	opprring 14433	. . . 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 14429	. . . . . . . . 9 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
33	32	eleq2d 2308	. . . . . . . 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 2238	. . . . . . 7 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
37		eqid 2238	. . . . . . 7 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
38		eqid 2238	. . . . . . 7 |- ( invg ` (oppr ` R ) ) = ( invg ` (oppr ` R ) )
39	36, 37, 38	dvdsrneg 14459	. . . . . 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 603	. . . . 5 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) ( ( invg ` (oppr ` R ) ) ` ( N ` X ) ) )
41	29, 11	opprnegg 14438	. . . . . . . 8 |- ( R e. Ring -> N = ( invg ` (oppr ` R ) ) )
42	41	fveq1d 5697	. . . . . . 7 |- ( R e. Ring -> ( N ` ( N ` X ) ) = ( ( invg ` (oppr ` R ) ) ` ( N ` X ) ) )
43	42	breq2d 4142	. . . . . 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 4156	. . 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 14457	. . 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 1278	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
50	5, 20, 21, 22, 23, 7	isunitd 14462	. 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 957	1 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) e. U )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377   Basecbs 13401   Grpcgrp 13854   invgcminusg 13855   1rcur 14311  SRingcsrg 14316   Ringcrg 14349  opp_rcoppr 14421   ||_rcdsr 14441  Unitcui 14442

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-tpos 6516  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9307  df-2 9365  df-3 9366  df-ndx 13404  df-slot 13405  df-base 13407  df-sets 13408  df-plusg 13493  df-mulr 13494  df-0g 13661  df-mgm 13725  df-sgrp 13766  df-mnd 13779  df-grp 13857  df-minusg 13858  df-cmn 14138  df-abl 14139  df-mgp 14267  df-ur 14312  df-srg 14317  df-ring 14351  df-oppr 14422  df-dvdsr 14444  df-unit 14445

This theorem is used by:  aprsym  14645