Theorem crngunit 14115

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 2230	. . . 4 ⊢ (𝑅 ∈ CRing → (oppr‘𝑅) = (oppr‘𝑅))
8		eqidd 2230	. . . 4 ⊢ (𝑅 ∈ CRing → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
9		crngring 14011	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
10		ringsrg 14050	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
11	9, 10	syl 14	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing)
12	2, 4, 6, 7, 8, 11	isunitd 14110	. . 3 ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋(∥r‘(oppr‘𝑅)) 1 )))
13		eqid 2229	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
14		eqid 2229	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
15		eqid 2229	. . . . . . . . . . . 12 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
16		eqid 2229	. . . . . . . . . . . 12 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
17	13, 14, 15, 16	crngoppr 14075	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑋) = (𝑦(.r‘(oppr‘𝑅))𝑋))
18	17	3expa 1227	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑋) = (𝑦(.r‘(oppr‘𝑅))𝑋))
19	18	eqcomd 2235	. . . . . . . . 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 2238	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r‘(oppr‘𝑅))𝑋) = 1 ↔ (𝑦(.r‘𝑅)𝑋) = 1 ))
22	21	rexbidva 2527	. . . . . 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 14078	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
25	15	opprring 14082	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
26		ringsrg 14050	. . . . . . 7 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ SRing)
27	9, 25, 26	3syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → (oppr‘𝑅) ∈ SRing)
28		eqidd 2230	. . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)))
29	24, 8, 27, 28	dvdsrd 14098	. . . . 5 ⊢ (𝑅 ∈ CRing → (𝑋(∥r‘(oppr‘𝑅)) 1 ↔ (𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑅))𝑋) = 1 )))
30		eqidd 2230	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
31		eqidd 2230	. . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘𝑅) = (.r‘𝑅))
32	30, 6, 11, 31	dvdsrd 14098	. . . . 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 1395   ∈ wcel 2200  ∃wrex 2509   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013  Basecbs 13072  ._rcmulr 13151  1_rcur 13962  SRingcsrg 13966  Ringcrg 13999  CRingccrg 14000  opp_rcoppr 14070  ∥_rcdsr 14089  Unitcui 14090

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-tpos 6406  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-plusg 13163  df-mulr 13164  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-minusg 13577  df-cmn 13863  df-abl 13864  df-mgp 13924  df-ur 13963  df-srg 13967  df-ring 14001  df-cring 14002  df-oppr 14071  df-dvdsr 14092  df-unit 14093

This theorem is referenced by:  dvdsunit  14116  cnfldui  14593  znunit  14663