Theorem crngunit 13667

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 2197	. . . 4 ⊢ (𝑅 ∈ CRing → (oppr‘𝑅) = (oppr‘𝑅))
8		eqidd 2197	. . . 4 ⊢ (𝑅 ∈ CRing → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
9		crngring 13564	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
10		ringsrg 13603	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
11	9, 10	syl 14	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing)
12	2, 4, 6, 7, 8, 11	isunitd 13662	. . 3 ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋(∥r‘(oppr‘𝑅)) 1 )))
13		eqid 2196	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
14		eqid 2196	. . . . . . . . . . . 12 ⊢ (.r‘𝑅) = (.r‘𝑅)
15		eqid 2196	. . . . . . . . . . . 12 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
16		eqid 2196	. . . . . . . . . . . 12 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
17	13, 14, 15, 16	crngoppr 13628	. . . . . . . . . . 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 2202	. . . . . . . . 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 2205	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r‘(oppr‘𝑅))𝑋) = 1 ↔ (𝑦(.r‘𝑅)𝑋) = 1 ))
22	21	rexbidva 2494	. . . . . 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 13631	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
25	15	opprring 13635	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
26		ringsrg 13603	. . . . . . 7 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ SRing)
27	9, 25, 26	3syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → (oppr‘𝑅) ∈ SRing)
28		eqidd 2197	. . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)))
29	24, 8, 27, 28	dvdsrd 13650	. . . . 5 ⊢ (𝑅 ∈ CRing → (𝑋(∥r‘(oppr‘𝑅)) 1 ↔ (𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑅))𝑋) = 1 )))
30		eqidd 2197	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
31		eqidd 2197	. . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘𝑅) = (.r‘𝑅))
32	30, 6, 11, 31	dvdsrd 13650	. . . . 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 2167  ∃wrex 2476   class class class wbr 4033  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  ._rcmulr 12756  1_rcur 13515  SRingcsrg 13519  Ringcrg 13552  CRingccrg 13553  opp_rcoppr 13623  ∥_rcdsr 13642  Unitcui 13643

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-tpos 6303  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-cmn 13416  df-abl 13417  df-mgp 13477  df-ur 13516  df-srg 13520  df-ring 13554  df-cring 13555  df-oppr 13624  df-dvdsr 13645  df-unit 13646

This theorem is referenced by:  dvdsunit  13668  cnfldui  14145  znunit  14215