Theorem isunitd 14064

Description: Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.)

Hypotheses

Ref	Expression
isunitd.1	|- ( ph -> U = (Unit ` R ) )
isunitd.2	|- ( ph -> .1. = ( 1r ` R ) )
isunitd.3	|- ( ph -> .|| = ( ||r ` R ) )
isunitd.4	|- ( ph -> S = (oppr ` R ) )
isunitd.5	|- ( ph -> E = ( ||r ` S ) )
isunitd.r	|- ( ph -> R e. SRing )

Assertion

Ref	Expression
isunitd	|- ( ph -> ( X e. U <-> ( X .|| .1. /\ X E .1. ) ) )

Proof of Theorem isunitd

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isunitd.1	. . . 4 |- ( ph -> U = (Unit ` R ) )
2		df-unit 14048	. . . . 5 |- Unit = ( r e. _V |-> ( `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) " { ( 1r ` r ) } ) )
3		fveq2 5626	. . . . . . . 8 |- ( r = R -> ( ||r ` r ) = ( ||r ` R ) )
4		2fveq3 5631	. . . . . . . 8 |- ( r = R -> ( ||r ` (oppr ` r ) ) = ( ||r ` (oppr ` R ) ) )
5	3, 4	ineq12d 3406	. . . . . . 7 |- ( r = R -> ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
6	5	cnveqd 4897	. . . . . 6 |- ( r = R -> `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
7		fveq2 5626	. . . . . . 7 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
8	7	sneqd 3679	. . . . . 6 |- ( r = R -> { ( 1r ` r ) } = { ( 1r ` R ) } )
9	6, 8	imaeq12d 5068	. . . . 5 |- ( r = R -> ( `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) " { ( 1r ` r ) } ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
10		isunitd.r	. . . . . 6 |- ( ph -> R e. SRing )
11	10	elexd 2813	. . . . 5 |- ( ph -> R e. _V )
12		dvdsrex 14056	. . . . . . 7 |- ( R e. SRing -> ( ||r ` R ) e. _V )
13		inex1g 4219	. . . . . . 7 |- ( ( ||r ` R ) e. _V -> ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V )
14	10, 12, 13	3syl 17	. . . . . 6 |- ( ph -> ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V )
15		cnvexg 5265	. . . . . 6 |- ( ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V -> `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V )
16		imaexg 5081	. . . . . 6 |- ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V -> ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) e. _V )
17	14, 15, 16	3syl 17	. . . . 5 |- ( ph -> ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) e. _V )
18	2, 9, 11, 17	fvmptd3 5727	. . . 4 |- ( ph -> (Unit ` R ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
19	1, 18	eqtrd 2262	. . 3 |- ( ph -> U = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
20	19	eleq2d 2299	. 2 |- ( ph -> ( X e. U <-> X e. ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) ) )
21		isunitd.3	. . . . . 6 |- ( ph -> .|| = ( ||r ` R ) )
22		isunitd.5	. . . . . . 7 |- ( ph -> E = ( ||r ` S ) )
23		isunitd.4	. . . . . . . 8 |- ( ph -> S = (oppr ` R ) )
24	23	fveq2d 5630	. . . . . . 7 |- ( ph -> ( ||r ` S ) = ( ||r ` (oppr ` R ) ) )
25	22, 24	eqtrd 2262	. . . . . 6 |- ( ph -> E = ( ||r ` (oppr ` R ) ) )
26	21, 25	ineq12d 3406	. . . . 5 |- ( ph -> ( .|| i^i E ) = ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
27	26	cnveqd 4897	. . . 4 |- ( ph -> `' ( .|| i^i E ) = `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
28		isunitd.2	. . . . 5 |- ( ph -> .1. = ( 1r ` R ) )
29	28	sneqd 3679	. . . 4 |- ( ph -> { .1. } = { ( 1r ` R ) } )
30	27, 29	imaeq12d 5068	. . 3 |- ( ph -> ( `' ( .|| i^i E ) " { .1. } ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
31	30	eleq2d 2299	. 2 |- ( ph -> ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> X e. ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) ) )
32		reldvdsrsrg 14050	. . . . . 6 |- ( R e. SRing -> Rel ( ||r ` R ) )
33	10, 32	syl 14	. . . . 5 |- ( ph -> Rel ( ||r ` R ) )
34	21	releqd 4802	. . . . 5 |- ( ph -> ( Rel .|| <-> Rel ( ||r ` R ) ) )
35	33, 34	mpbird 167	. . . 4 |- ( ph -> Rel .|| )
36		relin1 4836	. . . 4 |- ( Rel .|| -> Rel ( .|| i^i E ) )
37		eliniseg2 5107	. . . 4 |- ( Rel ( .|| i^i E ) -> ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> X ( .|| i^i E ) .1. ) )
38	35, 36, 37	3syl 17	. . 3 |- ( ph -> ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> X ( .|| i^i E ) .1. ) )
39		brin 4135	. . 3 |- ( X ( .|| i^i E ) .1. <-> ( X .|| .1. /\ X E .1. ) )
40	38, 39	bitrdi 196	. 2 |- ( ph -> ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> ( X .|| .1. /\ X E .1. ) ) )
41	20, 31, 40	3bitr2d 216	1 |- ( ph -> ( X e. U <-> ( X .|| .1. /\ X E .1. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   {csn 3666   class class class wbr 4082   `'ccnv 4717   "cima 4721   Rel wrel 4723   ` cfv 5317   1rcur 13917  SRingcsrg 13921  opp_rcoppr 14025   ||_rcdsr 14044  Unitcui 14045

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-plusg 13118  df-mulr 13119  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-mgp 13879  df-srg 13922  df-dvdsr 14047  df-unit 14048

This theorem is referenced by:  1unit  14065  unitcld  14066  opprunitd  14068  crngunit  14069  unitmulcl  14071  unitgrp  14074  unitnegcl  14088  unitpropdg  14106  elrhmunit  14135  subrguss  14194  subrgunit  14197