Theorem crngunit 13280

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 2178	. . . 4 |- ( R e. CRing -> (oppr ` R ) = (oppr ` R ) )
8		eqidd 2178	. . . 4 |- ( R e. CRing -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
9		crngring 13191	. . . . 5 |- ( R e. CRing -> R e. Ring )
10		ringsrg 13224	. . . . 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 13275	. . 3 |- ( R e. CRing -> ( X e. U <-> ( X .|| .1. /\ X ( ||r ` (oppr ` R ) ) .1. ) ) )
13		eqid 2177	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` R )
14		eqid 2177	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
15		eqid 2177	. . . . . . . . . . . 12 |- (oppr ` R ) = (oppr ` R )
16		eqid 2177	. . . . . . . . . . . 12 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
17	13, 14, 15, 16	crngoppr 13244	. . . . . . . . . . 11 |- ( ( R e. CRing /\ y e. ( Base ` R ) /\ X e. ( Base ` R ) ) -> ( y ( .r ` R ) X ) = ( y ( .r ` (oppr ` R ) ) X ) )
18	17	3expa 1203	. . . . . . . . . 10 |- ( ( ( R e. CRing /\ y e. ( Base ` R ) ) /\ X e. ( Base ` R ) ) -> ( y ( .r ` R ) X ) = ( y ( .r ` (oppr ` R ) ) X ) )
19	18	eqcomd 2183	. . . . . . . . 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 2186	. . . . . . 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 2474	. . . . . 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 13247	. . . . . 6 |- ( R e. CRing -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
25	15	opprring 13249	. . . . . . 7 |- ( R e. Ring -> (oppr ` R ) e. Ring )
26		ringsrg 13224	. . . . . . 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 2178	. . . . . 6 |- ( R e. CRing -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
29	24, 8, 27, 28	dvdsrd 13263	. . . . 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 2178	. . . . . 6 |- ( R e. CRing -> ( Base ` R ) = ( Base ` R ) )
31		eqidd 2178	. . . . . 6 |- ( R e. CRing -> ( .r ` R ) = ( .r ` R ) )
32	30, 6, 11, 31	dvdsrd 13263	. . . . 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 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4004   ` cfv 5217  (class class class)co 5875   Basecbs 12462   .rcmulr 12537   1rcur 13142  SRingcsrg 13146   Ringcrg 13179   CRingccrg 13180  opp_rcoppr 13239   ||_rcdsr 13255  Unitcui 13256

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-pre-ltirr 7923  ax-pre-lttrn 7925  ax-pre-ltadd 7927

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-tpos 6246  df-pnf 7994  df-mnf 7995  df-ltxr 7997  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-sets 12469  df-plusg 12549  df-mulr 12550  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-cmn 13090  df-abl 13091  df-mgp 13131  df-ur 13143  df-srg 13147  df-ring 13181  df-cring 13182  df-oppr 13240  df-dvdsr 13258  df-unit 13259

This theorem is referenced by:  dvdsunit  13281