Theorem isunit 18657

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
unit.1	|- U = (Unit ` R )
unit.2	|- .1. = ( 1r ` R )
unit.3	|- .|| = ( ||r ` R )
unit.4	|- S = (oppr ` R )
unit.5	|- E = ( ||r ` S )

Assertion

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

Proof of Theorem isunit

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6220	. . . 4 |- ( X e. (Unit ` R ) -> R e. dom Unit )
2		unit.1	. . . 4 |- U = (Unit ` R )
3	1, 2	eleq2s 2719	. . 3 |- ( X e. U -> R e. dom Unit )
4	3	elexd 3214	. 2 |- ( X e. U -> R e. _V )
5		df-br 4654	. . . 4 |- ( X .|| .1. <-> <. X , .1. >. e. .|| )
6		elfvdm 6220	. . . . . 6 |- ( <. X , .1. >. e. ( ||r ` R ) -> R e. dom ||r )
7		unit.3	. . . . . 6 |- .|| = ( ||r ` R )
8	6, 7	eleq2s 2719	. . . . 5 |- ( <. X , .1. >. e. .|| -> R e. dom ||r )
9	8	elexd 3214	. . . 4 |- ( <. X , .1. >. e. .|| -> R e. _V )
10	5, 9	sylbi 207	. . 3 |- ( X .|| .1. -> R e. _V )
11	10	adantr 481	. 2 |- ( ( X .|| .1. /\ X E .1. ) -> R e. _V )
12		fveq2 6191	. . . . . . . . . 10 |- ( r = R -> ( ||r ` r ) = ( ||r ` R ) )
13	12, 7	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> ( ||r ` r ) = .|| )
14		fveq2 6191	. . . . . . . . . . . 12 |- ( r = R -> (oppr ` r ) = (oppr ` R ) )
15		unit.4	. . . . . . . . . . . 12 |- S = (oppr ` R )
16	14, 15	syl6eqr 2674	. . . . . . . . . . 11 |- ( r = R -> (oppr ` r ) = S )
17	16	fveq2d 6195	. . . . . . . . . 10 |- ( r = R -> ( ||r ` (oppr ` r ) ) = ( ||r ` S ) )
18		unit.5	. . . . . . . . . 10 |- E = ( ||r ` S )
19	17, 18	syl6eqr 2674	. . . . . . . . 9 |- ( r = R -> ( ||r ` (oppr ` r ) ) = E )
20	13, 19	ineq12d 3815	. . . . . . . 8 |- ( r = R -> ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = ( .|| i^i E ) )
21	20	cnveqd 5298	. . . . . . 7 |- ( r = R -> `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = `' ( .|| i^i E ) )
22		fveq2 6191	. . . . . . . . 9 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
23		unit.2	. . . . . . . . 9 |- .1. = ( 1r ` R )
24	22, 23	syl6eqr 2674	. . . . . . . 8 |- ( r = R -> ( 1r ` r ) = .1. )
25	24	sneqd 4189	. . . . . . 7 |- ( r = R -> { ( 1r ` r ) } = { .1. } )
26	21, 25	imaeq12d 5467	. . . . . 6 |- ( r = R -> ( `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) " { ( 1r ` r ) } ) = ( `' ( .|| i^i E ) " { .1. } ) )
27		df-unit 18642	. . . . . 6 |- Unit = ( r e. _V |-> ( `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) " { ( 1r ` r ) } ) )
28		fvex 6201	. . . . . . . . . 10 |- ( ||r ` R ) e. _V
29	7, 28	eqeltri 2697	. . . . . . . . 9 |- .|| e. _V
30	29	inex1 4799	. . . . . . . 8 |- ( .|| i^i E ) e. _V
31	30	cnvex 7113	. . . . . . 7 |- `' ( .|| i^i E ) e. _V
32	31	imaex 7104	. . . . . 6 |- ( `' ( .|| i^i E ) " { .1. } ) e. _V
33	26, 27, 32	fvmpt 6282	. . . . 5 |- ( R e. _V -> (Unit ` R ) = ( `' ( .|| i^i E ) " { .1. } ) )
34	2, 33	syl5eq 2668	. . . 4 |- ( R e. _V -> U = ( `' ( .|| i^i E ) " { .1. } ) )
35	34	eleq2d 2687	. . 3 |- ( R e. _V -> ( X e. U <-> X e. ( `' ( .|| i^i E ) " { .1. } ) ) )
36		inss1 3833	. . . . . 6 |- ( .|| i^i E ) C_ .||
37	7	reldvdsr 18644	. . . . . 6 |- Rel .||
38		relss 5206	. . . . . 6 |- ( ( .|| i^i E ) C_ .|| -> ( Rel .|| -> Rel ( .|| i^i E ) ) )
39	36, 37, 38	mp2 9	. . . . 5 |- Rel ( .|| i^i E )
40		eliniseg2 5505	. . . . 5 |- ( Rel ( .|| i^i E ) -> ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> X ( .|| i^i E ) .1. ) )
41	39, 40	ax-mp 5	. . . 4 |- ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> X ( .|| i^i E ) .1. )
42		brin 4704	. . . 4 |- ( X ( .|| i^i E ) .1. <-> ( X .|| .1. /\ X E .1. ) )
43	41, 42	bitri 264	. . 3 |- ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> ( X .|| .1. /\ X E .1. ) )
44	35, 43	syl6bb 276	. 2 |- ( R e. _V -> ( X e. U <-> ( X .|| .1. /\ X E .1. ) ) )
45	4, 11, 44	pm5.21nii 368	1 |- ( X e. U <-> ( X .|| .1. /\ X E .1. ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   <.cop 4183   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   "cima 5117   Rel wrel 5119   ` cfv 5888   1rcur 18501  opp_rcoppr 18622   ||_rcdsr 18638  Unitcui 18639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-dvdsr 18641  df-unit 18642

This theorem is referenced by:  1unit  18658  unitcl  18659  opprunit  18661  crngunit  18662  unitmulcl  18664  unitgrp  18667  unitnegcl  18681  unitpropd  18697  isdrng2  18757  subrguss  18795  subrgunit  18798  fidomndrng  19307  invrvald  20482  elrhmunit  29820