Theorem isunitd 13662

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 13646	. . . . 5 |- Unit = ( r e. _V |-> ( `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) " { ( 1r ` r ) } ) )
3		fveq2 5558	. . . . . . . 8 |- ( r = R -> ( ||r ` r ) = ( ||r ` R ) )
4		2fveq3 5563	. . . . . . . 8 |- ( r = R -> ( ||r ` (oppr ` r ) ) = ( ||r ` (oppr ` R ) ) )
5	3, 4	ineq12d 3365	. . . . . . 7 |- ( r = R -> ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
6	5	cnveqd 4842	. . . . . 6 |- ( r = R -> `' ( ( ||r ` r ) i^i ( ||r ` (oppr ` r ) ) ) = `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
7		fveq2 5558	. . . . . . 7 |- ( r = R -> ( 1r ` r ) = ( 1r ` R ) )
8	7	sneqd 3635	. . . . . 6 |- ( r = R -> { ( 1r ` r ) } = { ( 1r ` R ) } )
9	6, 8	imaeq12d 5010	. . . . 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 2776	. . . . 5 |- ( ph -> R e. _V )
12		dvdsrex 13654	. . . . . . 7 |- ( R e. SRing -> ( ||r ` R ) e. _V )
13		inex1g 4169	. . . . . . 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 5207	. . . . . 6 |- ( ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V -> `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) e. _V )
16		imaexg 5023	. . . . . 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 5655	. . . 4 |- ( ph -> (Unit ` R ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
19	1, 18	eqtrd 2229	. . 3 |- ( ph -> U = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
20	19	eleq2d 2266	. 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 5562	. . . . . . 7 |- ( ph -> ( ||r ` S ) = ( ||r ` (oppr ` R ) ) )
25	22, 24	eqtrd 2229	. . . . . 6 |- ( ph -> E = ( ||r ` (oppr ` R ) ) )
26	21, 25	ineq12d 3365	. . . . 5 |- ( ph -> ( .|| i^i E ) = ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
27	26	cnveqd 4842	. . . 4 |- ( ph -> `' ( .|| i^i E ) = `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) )
28		isunitd.2	. . . . 5 |- ( ph -> .1. = ( 1r ` R ) )
29	28	sneqd 3635	. . . 4 |- ( ph -> { .1. } = { ( 1r ` R ) } )
30	27, 29	imaeq12d 5010	. . 3 |- ( ph -> ( `' ( .|| i^i E ) " { .1. } ) = ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) )
31	30	eleq2d 2266	. 2 |- ( ph -> ( X e. ( `' ( .|| i^i E ) " { .1. } ) <-> X e. ( `' ( ( ||r ` R ) i^i ( ||r ` (oppr ` R ) ) ) " { ( 1r ` R ) } ) ) )
32		reldvdsrsrg 13648	. . . . . 6 |- ( R e. SRing -> Rel ( ||r ` R ) )
33	10, 32	syl 14	. . . . 5 |- ( ph -> Rel ( ||r ` R ) )
34	21	releqd 4747	. . . . 5 |- ( ph -> ( Rel .|| <-> Rel ( ||r ` R ) ) )
35	33, 34	mpbird 167	. . . 4 |- ( ph -> Rel .|| )
36		relin1 4781	. . . 4 |- ( Rel .|| -> Rel ( .|| i^i E ) )
37		eliniseg2 5049	. . . 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 4085	. . 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 2167   _Vcvv 2763   i^i cin 3156   {csn 3622   class class class wbr 4033   `'ccnv 4662   "cima 4666   Rel wrel 4668   ` cfv 5258   1rcur 13515  SRingcsrg 13519  opp_rcoppr 13623   ||_rcdsr 13642  Unitcui 13643

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-mgp 13477  df-srg 13520  df-dvdsr 13645  df-unit 13646

This theorem is referenced by:  1unit  13663  unitcld  13664  opprunitd  13666  crngunit  13667  unitmulcl  13669  unitgrp  13672  unitnegcl  13686  unitpropdg  13704  elrhmunit  13733  subrguss  13792  subrgunit  13795