Theorem opprunitd 14114

Description: Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
opprunitd.1	⊢ (𝜑 → 𝑈 = (Unit‘𝑅))
opprunitd.2	⊢ (𝜑 → 𝑆 = (oppr‘𝑅))
opprunitd.r	⊢ (𝜑 → 𝑅 ∈ Ring)

Assertion

Ref	Expression
opprunitd	⊢ (𝜑 → 𝑈 = (Unit‘𝑆))

Proof of Theorem opprunitd

Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opprunitd.1	. . . . . 6 ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))
2		eqidd 2230	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑅))
3		eqidd 2230	. . . . . 6 ⊢ (𝜑 → (∥r‘𝑅) = (∥r‘𝑅))
4		opprunitd.2	. . . . . 6 ⊢ (𝜑 → 𝑆 = (oppr‘𝑅))
5		eqidd 2230	. . . . . 6 ⊢ (𝜑 → (∥r‘𝑆) = (∥r‘𝑆))
6		opprunitd.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		ringsrg 14050	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
8	6, 7	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ SRing)
9	1, 2, 3, 4, 5, 8	isunitd 14110	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘𝑆)(1r‘𝑅))))
10		eqid 2229	. . . . . . . . . . . . . . 15 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
11	10	opprring 14082	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
12	6, 11	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → (oppr‘𝑅) ∈ Ring)
13	4, 12	eqeltrd 2306	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Ring)
14		vex 2803	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
15	14	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑦 ∈ V)
16		vex 2803	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
17	16	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑥 ∈ V)
18		eqid 2229	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑆)
19		eqid 2229	. . . . . . . . . . . . 13 ⊢ (.r‘𝑆) = (.r‘𝑆)
20		eqid 2229	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
21		eqid 2229	. . . . . . . . . . . . 13 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
22	18, 19, 20, 21	opprmulg 14074	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Ring ∧ 𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(.r‘(oppr‘𝑆))𝑥) = (𝑥(.r‘𝑆)𝑦))
23	13, 15, 17, 22	syl3anc 1271	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦(.r‘(oppr‘𝑆))𝑥) = (𝑥(.r‘𝑆)𝑦))
24	4	fveq2d 5639	. . . . . . . . . . . 12 ⊢ (𝜑 → (.r‘𝑆) = (.r‘(oppr‘𝑅)))
25	24	oveqd 6030	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥(.r‘𝑆)𝑦) = (𝑥(.r‘(oppr‘𝑅))𝑦))
26		eqid 2229	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
27		eqid 2229	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
28		eqid 2229	. . . . . . . . . . . . 13 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
29	26, 27, 10, 28	opprmulg 14074	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
30	6, 17, 15, 29	syl3anc 1271	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
31	23, 25, 30	3eqtrrd 2267	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦(.r‘𝑅)𝑥) = (𝑦(.r‘(oppr‘𝑆))𝑥))
32	31	eqeq1d 2238	. . . . . . . . 9 ⊢ (𝜑 → ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ↔ (𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅)))
33	32	rexbidv 2531	. . . . . . . 8 ⊢ (𝜑 → (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅)))
34	33	anbi2d 464	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅)) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅))))
35		eqidd 2230	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
36		eqidd 2230	. . . . . . . 8 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑅))
37	35, 3, 8, 36	dvdsrd 14098	. . . . . . 7 ⊢ (𝜑 → (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))))
38	10, 26	opprbasg 14078	. . . . . . . . . 10 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
39	8, 38	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
40	4	fveq2d 5639	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(oppr‘𝑅)))
41	20, 18	opprbasg 14078	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → (Base‘𝑆) = (Base‘(oppr‘𝑆)))
42	13, 41	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(oppr‘𝑆)))
43	39, 40, 42	3eqtr2d 2268	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(oppr‘𝑆)))
44		eqidd 2230	. . . . . . . 8 ⊢ (𝜑 → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
45	20	opprring 14082	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → (oppr‘𝑆) ∈ Ring)
46	13, 45	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (oppr‘𝑆) ∈ Ring)
47		ringsrg 14050	. . . . . . . . 9 ⊢ ((oppr‘𝑆) ∈ Ring → (oppr‘𝑆) ∈ SRing)
48	46, 47	syl 14	. . . . . . . 8 ⊢ (𝜑 → (oppr‘𝑆) ∈ SRing)
49		eqidd 2230	. . . . . . . 8 ⊢ (𝜑 → (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆)))
50	43, 44, 48, 49	dvdsrd 14098	. . . . . . 7 ⊢ (𝜑 → (𝑥(∥r‘(oppr‘𝑆))(1r‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅))))
51	34, 37, 50	3bitr4d 220	. . . . . 6 ⊢ (𝜑 → (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑅)))
52	51	anbi1d 465	. . . . 5 ⊢ (𝜑 → ((𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘𝑆)(1r‘𝑅)) ↔ (𝑥(∥r‘(oppr‘𝑆))(1r‘𝑅) ∧ 𝑥(∥r‘𝑆)(1r‘𝑅))))
53	9, 52	bitrd 188	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘(oppr‘𝑆))(1r‘𝑅) ∧ 𝑥(∥r‘𝑆)(1r‘𝑅))))
54	53	biancomd 271	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑆)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑅))))
55		eqidd 2230	. . . 4 ⊢ (𝜑 → (Unit‘𝑆) = (Unit‘𝑆))
56		eqid 2229	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
57	10, 56	oppr1g 14085	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘(oppr‘𝑅)))
58	6, 57	syl 14	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) = (1r‘(oppr‘𝑅)))
59	4	fveq2d 5639	. . . . 5 ⊢ (𝜑 → (1r‘𝑆) = (1r‘(oppr‘𝑅)))
60	58, 59	eqtr4d 2265	. . . 4 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑆))
61		eqidd 2230	. . . 4 ⊢ (𝜑 → (oppr‘𝑆) = (oppr‘𝑆))
62		ringsrg 14050	. . . . 5 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
63	13, 62	syl 14	. . . 4 ⊢ (𝜑 → 𝑆 ∈ SRing)
64	55, 60, 5, 61, 44, 63	isunitd 14110	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Unit‘𝑆) ↔ (𝑥(∥r‘𝑆)(1r‘𝑅) ∧ 𝑥(∥r‘(oppr‘𝑆))(1r‘𝑅))))
65	54, 64	bitr4d 191	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↔ 𝑥 ∈ (Unit‘𝑆)))
66	65	eqrdv 2227	1 ⊢ (𝜑 → 𝑈 = (Unit‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  Vcvv 2800   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013  Basecbs 13072  ._rcmulr 13151  1_rcur 13962  SRingcsrg 13966  Ringcrg 13999  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-oppr 14071  df-dvdsr 14092  df-unit 14093

This theorem is referenced by: (None)