Theorem unitnegcl 14375

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 14244	. . . . . 6 |- ( R e. Ring -> R e. Grp )
3		eqidd 2235	. . . . . . 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 14290	. . . . . . . 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 14353	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> X e. ( Base ` R ) )
10		eqid 2234	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
11		unitnegcl.2	. . . . . . 7 |- N = ( invg ` R )
12	10, 11	grpinvcl 13803	. . . . . 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 2234	. . . . . 6 |- ( ||r ` R ) = ( ||r ` R )
15	10, 14, 11	dvdsrneg 14348	. . . . 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 13822	. . . . 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 4140	. . 3 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` R ) X )
20		eqidd 2235	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
21		eqidd 2235	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
22		eqidd 2235	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> (oppr ` R ) = (oppr ` R ) )
23		eqidd 2235	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
24	5, 20, 21, 22, 23, 7	isunitd 14351	. . . . 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 14346	. . 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 2234	. . . . 5 |- (oppr ` R ) = (oppr ` R )
30	29	opprring 14322	. . . 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 14318	. . . . . . . . 9 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
33	32	eleq2d 2304	. . . . . . . 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 2234	. . . . . . 7 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
37		eqid 2234	. . . . . . 7 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
38		eqid 2234	. . . . . . 7 |- ( invg ` (oppr ` R ) ) = ( invg ` (oppr ` R ) )
39	36, 37, 38	dvdsrneg 14348	. . . . . 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 14327	. . . . . . . 8 |- ( R e. Ring -> N = ( invg ` (oppr ` R ) ) )
42	41	fveq1d 5677	. . . . . . 7 |- ( R e. Ring -> ( N ` ( N ` X ) ) = ( ( invg ` (oppr ` R ) ) ` ( N ` X ) ) )
43	42	breq2d 4126	. . . . . 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 4140	. . 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 14346	. . 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 14351	. 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 2205   class class class wbr 4114   ` cfv 5357   Basecbs 13296   Grpcgrp 13755   invgcminusg 13756   1rcur 14202  SRingcsrg 14206   Ringcrg 14239  opp_rcoppr 14310   ||_rcdsr 14330  Unitcui 14331

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-tpos 6489  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-cmn 14039  df-abl 14040  df-mgp 14160  df-ur 14203  df-srg 14207  df-ring 14241  df-oppr 14311  df-dvdsr 14333  df-unit 14334

This theorem is referenced by:  aprsym  14534