Theorem unitnegcl 13834

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 13705	. . . . . 6 |- ( R e. Ring -> R e. Grp )
3		eqidd 2205	. . . . . . 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 13751	. . . . . . . 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 13812	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> X e. ( Base ` R ) )
10		eqid 2204	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
11		unitnegcl.2	. . . . . . 7 |- N = ( invg ` R )
12	10, 11	grpinvcl 13322	. . . . . 6 |- ( ( R e. Grp /\ X e. ( Base ` R ) ) -> ( N ` X ) e. ( Base ` R ) )
13	2, 9, 12	syl2an2r 595	. . . . 5 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) e. ( Base ` R ) )
14		eqid 2204	. . . . . 6 |- ( ||r ` R ) = ( ||r ` R )
15	10, 14, 11	dvdsrneg 13807	. . . . 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 13341	. . . . 5 |- ( ( R e. Grp /\ X e. ( Base ` R ) ) -> ( N ` ( N ` X ) ) = X )
18	2, 9, 17	syl2an2r 595	. . . 4 |- ( ( R e. Ring /\ X e. U ) -> ( N ` ( N ` X ) ) = X )
19	16, 18	breqtrd 4069	. . 3 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` R ) X )
20		eqidd 2205	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( 1r ` R ) = ( 1r ` R ) )
21		eqidd 2205	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ||r ` R ) = ( ||r ` R ) )
22		eqidd 2205	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> (oppr ` R ) = (oppr ` R ) )
23		eqidd 2205	. . . . . 6 |- ( ( R e. Ring /\ X e. U ) -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
24	5, 20, 21, 22, 23, 7	isunitd 13810	. . . . 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 13805	. . 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 1249	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` R ) ( 1r ` R ) )
29		eqid 2204	. . . . 5 |- (oppr ` R ) = (oppr ` R )
30	29	opprring 13783	. . . 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 13779	. . . . . . . . 9 |- ( R e. Ring -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
33	32	eleq2d 2274	. . . . . . . 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 2204	. . . . . . 7 |- ( Base ` (oppr ` R ) ) = ( Base ` (oppr ` R ) )
37		eqid 2204	. . . . . . 7 |- ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) )
38		eqid 2204	. . . . . . 7 |- ( invg ` (oppr ` R ) ) = ( invg ` (oppr ` R ) )
39	36, 37, 38	dvdsrneg 13807	. . . . . 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 595	. . . . 5 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) ( ( invg ` (oppr ` R ) ) ` ( N ` X ) ) )
41	29, 11	opprnegg 13787	. . . . . . . 8 |- ( R e. Ring -> N = ( invg ` (oppr ` R ) ) )
42	41	fveq1d 5577	. . . . . . 7 |- ( R e. Ring -> ( N ` ( N ` X ) ) = ( ( invg ` (oppr ` R ) ) ` ( N ` X ) ) )
43	42	breq2d 4055	. . . . . 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 4069	. . 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 13805	. . 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 1249	. 2 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) ( ||r ` (oppr ` R ) ) ( 1r ` R ) )
50	5, 20, 21, 22, 23, 7	isunitd 13810	. 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 946	1 |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) e. U )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   class class class wbr 4043   ` cfv 5270   Basecbs 12774   Grpcgrp 13274   invgcminusg 13275   1rcur 13663  SRingcsrg 13667   Ringcrg 13700  opp_rcoppr 13771   ||_rcdsr 13790  Unitcui 13791

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-tpos 6330  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12777  df-slot 12778  df-base 12780  df-sets 12781  df-plusg 12864  df-mulr 12865  df-0g 13032  df-mgm 13130  df-sgrp 13176  df-mnd 13191  df-grp 13277  df-minusg 13278  df-cmn 13564  df-abl 13565  df-mgp 13625  df-ur 13664  df-srg 13668  df-ring 13702  df-oppr 13772  df-dvdsr 13793  df-unit 13794

This theorem is referenced by:  aprsym  13988