Theorem isunitd 14413

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 14396	. . . . 5 |- Unit = ( r e. _V |-> ( `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) " { ( 1r ` r ) } ) )
3		fveq2 5695	. . . . . . . 8 |- ( r = R -> ( ||r ` r ) = ( ||r ` R ) )
4		2fveq3 5700	. . . . . . . 8 |- ( r = R -> ( ||r ` (oppr ` r ) ) = ( ||r ` (oppr ` R ) ) )
5	3, 4	ineq12d 3433	. . . . . . 7 |- ( r = R -> ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
6	5	cnveqd 4956	. . . . . 6 |- ( r = R -> `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
7		fveq2 5695	. . . . . . 7 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
8	7	sneqd 3722	. . . . . 6 |- ( r = R -> { ( 1r ` r ) } = { ( 1r ` R ) } )
9	6, 8	imaeq12d 5127	. . . . 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 2835	. . . . 5 |- ( ph -> R e. _V )
12		dvdsrex 14405	. . . . . . 7 |- ( R e. SRing -> ( ||r ` R ) e. _V )
13		inex1g 4269	. . . . . . 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 5325	. . . . . 6 |- ( ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V -> `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V )
16		imaexg 5140	. . . . . 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 5799	. . . 4 |- ( ph -> (Unit ` R ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
19	1, 18	eqtrd 2271	. . 3 |- ( ph -> U = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
20	19	eleq2d 2308	. 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 5699	. . . . . . 7 |- ( ph -> ( ||r ` S ) = ( ||r ` (oppr ` R ) ) )
25	22, 24	eqtrd 2271	. . . . . 6 |- ( ph -> E = ( ||r ` (oppr ` R ) ) )
26	21, 25	ineq12d 3433	. . . . 5 |- ( ph -> ( .|| i^i E ) = ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
27	26	cnveqd 4956	. . . 4 |- ( ph -> `' ( .|| i^i E ) = `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
28		isunitd.2	. . . . 5 |- ( ph -> .1. = ( 1r ` R ) )
29	28	sneqd 3722	. . . 4 |- ( ph -> { .1. } = { ( 1r ` R ) } )
30	27, 29	imaeq12d 5127	. . 3 |- ( ph -> ( `' ( .|| i^i E ) " { .1. } ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
31	30	eleq2d 2308	. 2 |- ( ph -> ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> X e. ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) ) )
32		reldvdsrsrg 14399	. . . . . 6 |- ( R e. SRing -> Rel ( ||r ` R ) )
33	10, 32	syl 14	. . . . 5 |- ( ph -> Rel ( ||r ` R ) )
34	21	releqd 4859	. . . . 5 |- ( ph -> ( Rel .|| <-> Rel ( ||r ` R ) ) )
35	33, 34	mpbird 167	. . . 4 |- ( ph -> Rel .|| )
36		relin1 4895	. . . 4 |- ( Rel .|| -> Rel ( .|| i^i E ) )
37		eliniseg2 5167	. . . 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 4183	. . 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. ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   {csn 3709   class class class wbr 4130   `'ccnv 4773   "cima 4777   Rel wrel 4779   ` cfv 5377   1rcur 14262  SRingcsrg 14267  opp_rcoppr 14372   ||_rcdsr 14392  Unitcui 14393

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-sep 4249  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-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-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-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-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-mgp 14218  df-srg 14268  df-dvdsr 14395  df-unit 14396

This theorem is used by:  1unit  14414  unitcld  14415  opprunitd  14417  crngunit  14418  unitmulcl  14420  unitgrp  14423  unitnegcl  14437  unitpropdg  14455  elrhmunit  14484  subrguss  14544  subrgunit  14547