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 ∈ 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 2239	. . . 4 ⊢ (𝑅 ∈ CRing → (oppr‘𝑅) = (oppr‘𝑅))
8		eqidd 2239	. . . 4 ⊢ (𝑅 ∈ CRing → (∥r‘(oppr‘𝑅)) = (∥r‘(oppr‘𝑅)))
9		crngring 14419	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
10		ringsrg 14459	. . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
11	9, 10	syl 14	. . . 4 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ SRing)
12	2, 4, 6, 7, 8, 11	isunitd 14522	. . 3 ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 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 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑋) = (𝑦(.r‘(oppr‘𝑅))𝑋))
18	17	3expa 1234	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘𝑅)𝑋) = (𝑦(.r‘(oppr‘𝑅))𝑋))
19	18	eqcomd 2244	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑦(.r‘(oppr‘𝑅))𝑋) = (𝑦(.r‘𝑅)𝑋))
20	19	an32s 574	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r‘(oppr‘𝑅))𝑋) = (𝑦(.r‘𝑅)𝑋))
21	20	eqeq1d 2247	. . . . . . 7 ⊢ (((𝑅 ∈ CRing ∧ 𝑋 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r‘(oppr‘𝑅))𝑋) = 1 ↔ (𝑦(.r‘𝑅)𝑋) = 1 ))
22	21	rexbidva 2547	. . . . . 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 456	. . . . 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 14489	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
25	15	opprring 14493	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
26		ringsrg 14459	. . . . . . 7 ⊢ ((oppr‘𝑅) ∈ Ring → (oppr‘𝑅) ∈ SRing)
27	9, 25, 26	3syl 17	. . . . . 6 ⊢ (𝑅 ∈ CRing → (oppr‘𝑅) ∈ SRing)
28		eqidd 2239	. . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)))
29	24, 8, 27, 28	dvdsrd 14510	. . . . 5 ⊢ (𝑅 ∈ CRing → (𝑋(∥r‘(oppr‘𝑅)) 1 ↔ (𝑋 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑅))𝑋) = 1 )))
30		eqidd 2239	. . . . . 6 ⊢ (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
31		eqidd 2239	. . . . . 6 ⊢ (𝑅 ∈ CRing → (.r‘𝑅) = (.r‘𝑅))
32	30, 6, 11, 31	dvdsrd 14510	. . . . 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 468	. . 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		pm4.24 399	. 2 ⊢ (𝑋 ∥ 1 ↔ (𝑋 ∥ 1 ∧ 𝑋 ∥ 1 ))
37	35, 36	bitr4di 198	1 ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∥ 1 ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∃wrex 2529   class class class wbr 4131  ‘cfv 5380  (class class class)co 6088  Basecbs 13410  ._rcmulr 13491  1_rcur 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