Theorem crngunit 13988

Description: Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)

Hypotheses

Ref	Expression
crngunit.1	|- U = (Unit ` R )
crngunit.2	|- .1. = ( 1r ` R )
crngunit.3	|- .|| = ( ||r ` R )

Assertion

Ref	Expression
crngunit	|- ( R e. CRing -> ( X e. U <-> X .|| .1. ) )

Proof of Theorem crngunit

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngunit.1	. . . . 5 |- U = (Unit ` R )
2	1	a1i 9	. . . 4 |- ( R e. CRing -> U = (Unit ` R ) )
3		crngunit.2	. . . . 5 |- .1. = ( 1r ` R )
4	3	a1i 9	. . . 4 |- ( R e. CRing -> .1. = ( 1r ` R ) )
5		crngunit.3	. . . . 5 |- .|| = ( ||r ` R )
6	5	a1i 9	. . . 4 |- ( R e. CRing -> .|| = ( ||r ` R ) )
7		eqidd 2208	. . . 4 |- ( R e. CRing -> (oppr ` R ) = (oppr ` R ) )
8		eqidd 2208	. . . 4 |- ( R e. CRing -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
9		crngring 13885	. . . . 5 |- ( R e. CRing -> R e. Ring )
10		ringsrg 13924	. . . . 5 |- ( R e. Ring -> R e. SRing )
11	9, 10	syl 14	. . . 4 |- ( R e. CRing -> R e. SRing )
12	2, 4, 6, 7, 8, 11	isunitd 13983	. . 3 |- ( R e. CRing -> ( X e. U <-> ( X .|| .1. /\ X ( ||r ` (oppr ` R ) ) .1. ) ) )
13		eqid 2207	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` R )
14		eqid 2207	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
15		eqid 2207	. . . . . . . . . . . 12 |- (oppr ` R ) = (oppr ` R )
16		eqid 2207	. . . . . . . . . . . 12 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
17	13, 14, 15, 16	crngoppr 13949	. . . . . . . . . . 11 |- ( ( R e. CRing /\ y e. ( Base ` R ) /\ X e. ( Base ` R ) ) -> ( y ( .r ` R ) X ) = ( y ( .r ` (oppr ` R ) ) X ) )
18	17	3expa 1206	. . . . . . . . . 10 |- ( ( ( R e. CRing /\ y e. ( Base ` R ) ) /\ X e. ( Base ` R ) ) -> ( y ( .r ` R ) X ) = ( y ( .r ` (oppr ` R ) ) X ) )
19	18	eqcomd 2213	. . . . . . . . 9 |- ( ( ( R e. CRing /\ y e. ( Base ` R ) ) /\ X e. ( Base ` R ) ) -> ( y ( .r ` (oppr ` R ) ) X ) = ( y ( .r ` R ) X ) )
20	19	an32s 568	. . . . . . . 8 |- ( ( ( R e. CRing /\ X e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( y ( .r ` (oppr ` R ) ) X ) = ( y ( .r ` R ) X ) )
21	20	eqeq1d 2216	. . . . . . 7 |- ( ( ( R e. CRing /\ X e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( ( y ( .r ` (oppr ` R ) ) X ) = .1. <-> ( y ( .r ` R ) X ) = .1. ) )
22	21	rexbidva 2505	. . . . . 6 |- ( ( R e. CRing /\ X e. ( Base ` R ) ) -> ( E. y e. ( Base ` R ) ( y ( .r ` (oppr ` R ) ) X ) = .1. <-> E. y e. ( Base ` R ) ( y ( .r ` R ) X ) = .1. ) )
23	22	pm5.32da 452	. . . . 5 |- ( R e. CRing -> ( ( X e. ( Base ` R ) /\ E. y e. ( Base ` R ) ( y ( .r ` (oppr ` R ) ) X ) = .1. ) <-> ( X e. ( Base ` R ) /\ E. y e. ( Base ` R ) ( y ( .r ` R ) X ) = .1. ) ) )
24	15, 13	opprbasg 13952	. . . . . 6 |- ( R e. CRing -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
25	15	opprring 13956	. . . . . . 7 |- ( R e. Ring -> (oppr ` R ) e. Ring )
26		ringsrg 13924	. . . . . . 7 |- ( (oppr ` R ) e. Ring -> (oppr ` R ) e. SRing )
27	9, 25, 26	3syl 17	. . . . . 6 |- ( R e. CRing -> (oppr ` R ) e. SRing )
28		eqidd 2208	. . . . . 6 |- ( R e. CRing -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
29	24, 8, 27, 28	dvdsrd 13971	. . . . 5 |- ( R e. CRing -> ( X ( ||r ` (oppr ` R ) ) .1. <-> ( X e. ( Base ` R ) /\ E. y e. ( Base ` R ) ( y ( .r ` (oppr ` R ) ) X ) = .1. ) ) )
30		eqidd 2208	. . . . . 6 |- ( R e. CRing -> ( Base ` R ) = ( Base ` R ) )
31		eqidd 2208	. . . . . 6 |- ( R e. CRing -> ( .r ` R ) = ( .r ` R ) )
32	30, 6, 11, 31	dvdsrd 13971	. . . . 5 |- ( R e. CRing -> ( X .|| .1. <-> ( X e. ( Base ` R ) /\ E. y e. ( Base ` R ) ( y ( .r ` R ) X ) = .1. ) ) )
33	23, 29, 32	3bitr4d 220	. . . 4 |- ( R e. CRing -> ( X ( ||r ` (oppr ` R ) ) .1. <-> X .|| .1. ) )
34	33	anbi2d 464	. . 3 |- ( R e. CRing -> ( ( X .|| .1. /\ X ( ||r ` (oppr ` R ) ) .1. ) <-> ( X .|| .1. /\ X .|| .1. ) ) )
35	12, 34	bitrd 188	. 2 |- ( R e. CRing -> ( X e. U <-> ( X .|| .1. /\ X .|| .1. ) ) )
36		pm4.24 395	. 2 |- ( X .|| .1. <-> ( X .|| .1. /\ X .|| .1. ) )
37	35, 36	bitr4di 198	1 |- ( R e. CRing -> ( X e. U <-> X .|| .1. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   E.wrex 2487   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   Basecbs 12947   .rcmulr 13025   1rcur 13836  SRingcsrg 13840   Ringcrg 13873   CRingccrg 13874  opp_rcoppr 13944   ||_rcdsr 13963  Unitcui 13964

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-tpos 6354  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-cmn 13737  df-abl 13738  df-mgp 13798  df-ur 13837  df-srg 13841  df-ring 13875  df-cring 13876  df-oppr 13945  df-dvdsr 13966  df-unit 13967

This theorem is referenced by:  dvdsunit  13989  cnfldui  14466  znunit  14536