Theorem crngunit 14527

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 2239	. . . 4 |- ( R e. CRing -> (oppr ` R ) = (oppr ` R ) )
8		eqidd 2239	. . . 4 |- ( R e. CRing -> ( ||r ` (oppr ` R ) ) = ( ||r ` (oppr ` R ) ) )
9		crngring 14419	. . . . 5 |- ( R e. CRing -> R e. Ring )
10		ringsrg 14459	. . . . 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 14522	. . 3 |- ( R e. CRing -> ( X e. U <-> ( X .|| .1. /\ X ( ||r ` (oppr ` R ) ) .1. ) ) )
13		eqid 2238	. . . . . . . . . . . 12 |- ( Base ` R ) = ( Base ` R )
14		eqid 2238	. . . . . . . . . . . 12 |- ( .r ` R ) = ( .r ` R )
15		eqid 2238	. . . . . . . . . . . 12 |- (oppr ` R ) = (oppr ` R )
16		eqid 2238	. . . . . . . . . . . 12 |- ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) )
17	13, 14, 15, 16	crngoppr 14486	. . . . . . . . . . 11 |- ( ( R e. CRing /\ y e. ( Base ` R ) /\ X e. ( Base ` R ) ) -> ( y ( .r ` R ) X ) = ( y ( .r ` (oppr ` R ) ) X ) )
18	17	3expa 1234	. . . . . . . . . 10 |- ( ( ( R e. CRing /\ y e. ( Base ` R ) ) /\ X e. ( Base ` R ) ) -> ( y ( .r ` R ) X ) = ( y ( .r ` (oppr ` R ) ) X ) )
19	18	eqcomd 2244	. . . . . . . . 9 |- ( ( ( R e. CRing /\ y e. ( Base ` R ) ) /\ X e. ( Base ` R ) ) -> ( y ( .r ` (oppr ` R ) ) X ) = ( y ( .r ` R ) X ) )
20	19	an32s 574	. . . . . . . 8 |- ( ( ( R e. CRing /\ X e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( y ( .r ` (oppr ` R ) ) X ) = ( y ( .r ` R ) X ) )
21	20	eqeq1d 2247	. . . . . . 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 2547	. . . . . 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 456	. . . . 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 14489	. . . . . 6 |- ( R e. CRing -> ( Base ` R ) = ( Base ` (oppr ` R ) ) )
25	15	opprring 14493	. . . . . . 7 |- ( R e. Ring -> (oppr ` R ) e. Ring )
26		ringsrg 14459	. . . . . . 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 2239	. . . . . 6 |- ( R e. CRing -> ( .r ` (oppr ` R ) ) = ( .r ` (oppr ` R ) ) )
29	24, 8, 27, 28	dvdsrd 14510	. . . . 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 2239	. . . . . 6 |- ( R e. CRing -> ( Base ` R ) = ( Base ` R ) )
31		eqidd 2239	. . . . . 6 |- ( R e. CRing -> ( .r ` R ) = ( .r ` R ) )
32	30, 6, 11, 31	dvdsrd 14510	. . . . 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 468	. . 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		pm4.24 399	. 2 |- ( X .|| .1. <-> ( X .|| .1. /\ X .|| .1. ) )
37	35, 36	bitr4di 198	1 |- ( R e. CRing -> ( X e. U <-> X .|| .1. ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   Basecbs 13410   .rcmulr 13491   1rcur 14369  SRingcsrg 14374   Ringcrg 14407   CRingccrg 14408  opp_rcoppr 14481   ||_rcdsr 14501  Unitcui 14502

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-tpos 6520  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-plusg 13503  df-mulr 13504  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-cmn 14179  df-abl 14180  df-mgp 14325  df-ur 14370  df-srg 14375  df-ring 14409  df-cring 14410  df-oppr 14482  df-dvdsr 14504  df-unit 14505

This theorem is used by:  dvdsunit  14528  cnfldui  15033  znunit  15103