Theorem isunitd 14251

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 14234	. . . . 5 |- Unit = ( r e. _V |-> ( `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) " { ( 1r ` r ) } ) )
3		fveq2 5670	. . . . . . . 8 |- ( r = R -> ( ||r ` r ) = ( ||r ` R ) )
4		2fveq3 5675	. . . . . . . 8 |- ( r = R -> ( ||r ` (oppr ` r ) ) = ( ||r ` (oppr ` R ) ) )
5	3, 4	ineq12d 3423	. . . . . . 7 |- ( r = R -> ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
6	5	cnveqd 4931	. . . . . 6 |- ( r = R -> `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
7		fveq2 5670	. . . . . . 7 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
8	7	sneqd 3702	. . . . . 6 |- ( r = R -> { ( 1r ` r ) } = { ( 1r ` R ) } )
9	6, 8	imaeq12d 5102	. . . . 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 2827	. . . . 5 |- ( ph -> R e. _V )
12		dvdsrex 14243	. . . . . . 7 |- ( R e. SRing -> ( ||r ` R ) e. _V )
13		inex1g 4246	. . . . . . 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 5300	. . . . . 6 |- ( ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V -> `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V )
16		imaexg 5115	. . . . . 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 5771	. . . 4 |- ( ph -> (Unit ` R ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
19	1, 18	eqtrd 2265	. . 3 |- ( ph -> U = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
20	19	eleq2d 2302	. 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 5674	. . . . . . 7 |- ( ph -> ( ||r ` S ) = ( ||r ` (oppr ` R ) ) )
25	22, 24	eqtrd 2265	. . . . . 6 |- ( ph -> E = ( ||r ` (oppr ` R ) ) )
26	21, 25	ineq12d 3423	. . . . 5 |- ( ph -> ( .|| i^i E ) = ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
27	26	cnveqd 4931	. . . 4 |- ( ph -> `' ( .|| i^i E ) = `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
28		isunitd.2	. . . . 5 |- ( ph -> .1. = ( 1r ` R ) )
29	28	sneqd 3702	. . . 4 |- ( ph -> { .1. } = { ( 1r ` R ) } )
30	27, 29	imaeq12d 5102	. . 3 |- ( ph -> ( `' ( .|| i^i E ) " { .1. } ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
31	30	eleq2d 2302	. 2 |- ( ph -> ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> X e. ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) ) )
32		reldvdsrsrg 14237	. . . . . 6 |- ( R e. SRing -> Rel ( ||r ` R ) )
33	10, 32	syl 14	. . . . 5 |- ( ph -> Rel ( ||r ` R ) )
34	21	releqd 4834	. . . . 5 |- ( ph -> ( Rel .|| <-> Rel ( ||r ` R ) ) )
35	33, 34	mpbird 167	. . . 4 |- ( ph -> Rel .|| )
36		relin1 4870	. . . 4 |- ( Rel .|| -> Rel ( .|| i^i E ) )
37		eliniseg2 5142	. . . 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 4162	. . 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 1398   e. wcel 2203   _Vcvv 2813   i^i cin 3210   {csn 3689   class class class wbr 4109   `'ccnv 4748   "cima 4752   Rel wrel 4754   ` cfv 5352   1rcur 14103  SRingcsrg 14107  opp_rcoppr 14211   ||_rcdsr 14230  Unitcui 14231

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-mgp 14065  df-srg 14108  df-dvdsr 14233  df-unit 14234

This theorem is referenced by:  1unit  14252  unitcld  14253  opprunitd  14255  crngunit  14256  unitmulcl  14258  unitgrp  14261  unitnegcl  14275  unitpropdg  14293  elrhmunit  14322  subrguss  14381  subrgunit  14384