Theorem isunitd 13605

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 13589	. . . . 5 |- Unit = ( r e. _V |-> ( `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) " { ( 1r ` r ) } ) )
3		fveq2 5555	. . . . . . . 8 |- ( r = R -> ( ||r ` r ) = ( ||r ` R ) )
4		2fveq3 5560	. . . . . . . 8 |- ( r = R -> ( ||r ` (oppr ` r ) ) = ( ||r ` (oppr ` R ) ) )
5	3, 4	ineq12d 3362	. . . . . . 7 |- ( r = R -> ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
6	5	cnveqd 4839	. . . . . 6 |- ( r = R -> `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
7		fveq2 5555	. . . . . . 7 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
8	7	sneqd 3632	. . . . . 6 |- ( r = R -> { ( 1r ` r ) } = { ( 1r ` R ) } )
9	6, 8	imaeq12d 5007	. . . . 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 2773	. . . . 5 |- ( ph -> R e. _V )
12		dvdsrex 13597	. . . . . . 7 |- ( R e. SRing -> ( ||r ` R ) e. _V )
13		inex1g 4166	. . . . . . 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 5204	. . . . . 6 |- ( ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V -> `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V )
16		imaexg 5020	. . . . . 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 5652	. . . 4 |- ( ph -> (Unit ` R ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
19	1, 18	eqtrd 2226	. . 3 |- ( ph -> U = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
20	19	eleq2d 2263	. 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 5559	. . . . . . 7 |- ( ph -> ( ||r ` S ) = ( ||r ` (oppr ` R ) ) )
25	22, 24	eqtrd 2226	. . . . . 6 |- ( ph -> E = ( ||r ` (oppr ` R ) ) )
26	21, 25	ineq12d 3362	. . . . 5 |- ( ph -> ( .|| i^i E ) = ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
27	26	cnveqd 4839	. . . 4 |- ( ph -> `' ( .|| i^i E ) = `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
28		isunitd.2	. . . . 5 |- ( ph -> .1. = ( 1r ` R ) )
29	28	sneqd 3632	. . . 4 |- ( ph -> { .1. } = { ( 1r ` R ) } )
30	27, 29	imaeq12d 5007	. . 3 |- ( ph -> ( `' ( .|| i^i E ) " { .1. } ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
31	30	eleq2d 2263	. 2 |- ( ph -> ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> X e. ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) ) )
32		reldvdsrsrg 13591	. . . . . 6 |- ( R e. SRing -> Rel ( ||r ` R ) )
33	10, 32	syl 14	. . . . 5 |- ( ph -> Rel ( ||r ` R ) )
34	21	releqd 4744	. . . . 5 |- ( ph -> ( Rel .|| <-> Rel ( ||r ` R ) ) )
35	33, 34	mpbird 167	. . . 4 |- ( ph -> Rel .|| )
36		relin1 4778	. . . 4 |- ( Rel .|| -> Rel ( .|| i^i E ) )
37		eliniseg2 5046	. . . 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 4082	. . 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 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3153   {csn 3619   class class class wbr 4030   `'ccnv 4659   "cima 4663   Rel wrel 4665   ` cfv 5255   1rcur 13458  SRingcsrg 13462  opp_rcoppr 13566   ||_rcdsr 13585  Unitcui 13586

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mgp 13420  df-srg 13463  df-dvdsr 13588  df-unit 13589

This theorem is referenced by:  1unit  13606  unitcld  13607  opprunitd  13609  crngunit  13610  unitmulcl  13612  unitgrp  13615  unitnegcl  13629  unitpropdg  13647  elrhmunit  13676  subrguss  13735  subrgunit  13738