Theorem crngunit 13607

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 ∈ CRing → (𝑋 ∈ 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 ∈ CRing → 𝑈 = (Unit‘𝑅))
3		crngunit.2	. . . . 5 ⊢ 1 = (1r‘𝑅)
4	3	a1i 9	. . . 4 ⊢ (𝑅 ∈ CRing → 1 = (1r‘𝑅))
5		crngunit.3	. . . . 5 ⊢ ∥ = (∥r‘𝑅)
6	5	a1i 9	. . . 4 ⊢ (𝑅 ∈ CRing → ∥ = (∥r‘𝑅))
7		eqidd 2194	. . . 4 ⊢ (𝑅 ∈ CRing → (oppr‘𝑅) = (oppr‘𝑅))
8		eqidd 2194	. . . 4 ⊢ (𝑅 ∈ CRing → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
9		crngring 13504	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
10		ringsrg 13543	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
11	9, 10	syl 14	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing)
12	2, 4, 6, 7, 8, 11	isunitd 13602	. . 3 ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋(∥r‘(oppr‘𝑅)) 1 )))
13		eqid 2193	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
14		eqid 2193	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
15		eqid 2193	. . . . . . . . . . . 12 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
16		eqid 2193	. . . . . . . . . . . 12 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
17	13, 14, 15, 16	crngoppr 13568	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑋) = (𝑦(.r‘(oppr‘𝑅))𝑋))
18	17	3expa 1205	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑋) = (𝑦(.r‘(oppr‘𝑅))𝑋))
19	18	eqcomd 2199	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘(oppr‘𝑅))𝑋) = (𝑦(.r‘𝑅)𝑋))
20	19	an32s 568	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘(oppr‘𝑅))𝑋) = (𝑦(.r‘𝑅)𝑋))
21	20	eqeq1d 2202	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r‘(oppr‘𝑅))𝑋) = 1 ↔ (𝑦(.r‘𝑅)𝑋) = 1 ))
22	21	rexbidva 2491	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘𝑅)) → (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑅))𝑋) = 1 ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑋) = 1 ))
23	22	pm5.32da 452	. . . . 5 ⊢ (𝑅 ∈ CRing → ((𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑅))𝑋) = 1 ) ↔ (𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑋) = 1 )))
24	15, 13	opprbasg 13571	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
25	15	opprring 13575	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
26		ringsrg 13543	. . . . . . 7 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ SRing)
27	9, 25, 26	3syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → (oppr‘𝑅) ∈ SRing)
28		eqidd 2194	. . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)))
29	24, 8, 27, 28	dvdsrd 13590	. . . . 5 ⊢ (𝑅 ∈ CRing → (𝑋(∥r‘(oppr‘𝑅)) 1 ↔ (𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑅))𝑋) = 1 )))
30		eqidd 2194	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
31		eqidd 2194	. . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘𝑅) = (.r‘𝑅))
32	30, 6, 11, 31	dvdsrd 13590	. . . . 5 ⊢ (𝑅 ∈ CRing → (𝑋 ∥ 1 ↔ (𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑋) = 1 )))
33	23, 29, 32	3bitr4d 220	. . . 4 ⊢ (𝑅 ∈ CRing → (𝑋(∥r‘(oppr‘𝑅)) 1 ↔ 𝑋 ∥ 1 ))
34	33	anbi2d 464	. . 3 ⊢ (𝑅 ∈ CRing → ((𝑋 ∥ 1 ∧ 𝑋(∥r‘(oppr‘𝑅)) 1 ) ↔ (𝑋 ∥ 1 ∧ 𝑋 ∥ 1 )))
35	12, 34	bitrd 188	. 2 ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋 ∥ 1 )))
36		pm4.24 395	. 2 ⊢ (𝑋 ∥ 1 ↔ (𝑋 ∥ 1 ∧ 𝑋 ∥ 1 ))
37	35, 36	bitr4di 198	1 ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∥ 1 ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∃wrex 2473   class class class wbr 4029  ‘cfv 5254  (class class class)co 5918  Basecbs 12618  ._rcmulr 12696  1_rcur 13455  SRingcsrg 13459  Ringcrg 13492  CRingccrg 13493  opp_rcoppr 13563  ∥_rcdsr 13582  Unitcui 13583

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-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-cmn 13356  df-abl 13357  df-mgp 13417  df-ur 13456  df-srg 13460  df-ring 13494  df-cring 13495  df-oppr 13564  df-dvdsr 13585  df-unit 13586

This theorem is referenced by:  dvdsunit  13608  cnfldui  14077  znunit  14147