Theorem unitnegcl 14143

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 14013	. . . . . 6 |- ( R e. Ring -> R e. Grp )
3		eqidd 2232	. . . . . . 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 14059	. . . . . . . 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 14121	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> X e. ( Base ` R ) )
10		eqid 2231	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
11		unitnegcl.2	. . . . . . 7 |- N = ( invg ` R )
12	10, 11	grpinvcl 13630	. . . . . 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 2231	. . . . . 6 |- ( ||r ` R ) = ( ||r ` R )
15	10, 14, 11	dvdsrneg 14116	. . . . 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 13649	. . . . 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 4114	. . 3 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` R ) X )
20		eqidd 2232	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
21		eqidd 2232	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
22		eqidd 2232	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> (oppr ` R ) = (oppr ` R ) )
23		eqidd 2232	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
24	5, 20, 21, 22, 23, 7	isunitd 14119	. . . . 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 14114	. . 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 1273	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` R ) ( 1r ` R ) )
29		eqid 2231	. . . . 5 |- (oppr ` R ) = (oppr ` R )
30	29	opprring 14091	. . . 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 14087	. . . . . . . . 9 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
33	32	eleq2d 2301	. . . . . . . 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 2231	. . . . . . 7 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
37		eqid 2231	. . . . . . 7 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
38		eqid 2231	. . . . . . 7 |- ( invg ` (oppr ` R ) ) = ( invg ` (oppr ` R ) )
39	36, 37, 38	dvdsrneg 14116	. . . . . 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 14095	. . . . . . . 8 |- ( R e. Ring -> N = ( invg ` (oppr ` R ) ) )
42	41	fveq1d 5641	. . . . . . 7 |- ( R e. Ring -> ( N ` ( N ` X ) ) = ( ( invg ` (oppr ` R ) ) ` ( N ` X ) ) )
43	42	breq2d 4100	. . . . . 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 4114	. . 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 14114	. . 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 1273	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
50	5, 20, 21, 22, 23, 7	isunitd 14119	. 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 952	1 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) e. U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326   Basecbs 13081   Grpcgrp 13582   invgcminusg 13583   1rcur 13971  SRingcsrg 13975   Ringcrg 14008  opp_rcoppr 14079   ||_rcdsr 14098  Unitcui 14099

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-cmn 13872  df-abl 13873  df-mgp 13933  df-ur 13972  df-srg 13976  df-ring 14010  df-oppr 14080  df-dvdsr 14101  df-unit 14102

This theorem is referenced by:  aprsym  14297