Theorem crngunit 13610

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 13507	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
10		ringsrg 13546	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
11	9, 10	syl 14	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing)
12	2, 4, 6, 7, 8, 11	isunitd 13605	. . 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 13571	. . . . . . . . . . 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 13574	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
25	15	opprring 13578	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
26		ringsrg 13546	. . . . . . 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 13593	. . . . 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 13593	. . . . 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 4030  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  ._rcmulr 12699  1_rcur 13458  SRingcsrg 13462  Ringcrg 13495  CRingccrg 13496  opp_rcoppr 13566  ∥_rcdsr 13585  Unitcui 13586

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 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

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 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-tpos 6300  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-cmn 13359  df-abl 13360  df-mgp 13420  df-ur 13459  df-srg 13463  df-ring 13497  df-cring 13498  df-oppr 13567  df-dvdsr 13588  df-unit 13589

This theorem is referenced by:  dvdsunit  13611  cnfldui  14088  znunit  14158