Theorem opprunitd 13341

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 2188	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑅))
3		eqidd 2188	. . . . . 6 ⊢ (𝜑 → (∥r‘𝑅) = (∥r‘𝑅))
4		opprunitd.2	. . . . . 6 ⊢ (𝜑 → 𝑆 = (oppr‘𝑅))
5		eqidd 2188	. . . . . 6 ⊢ (𝜑 → (∥r‘𝑆) = (∥r‘𝑆))
6		opprunitd.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		ringsrg 13282	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
8	6, 7	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ SRing)
9	1, 2, 3, 4, 5, 8	isunitd 13337	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘𝑆)(1r‘𝑅))))
10		eqid 2187	. . . . . . . . . . . . . . 15 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
11	10	opprring 13310	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
12	6, 11	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → (oppr‘𝑅) ∈ Ring)
13	4, 12	eqeltrd 2264	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Ring)
14		vex 2752	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
15	14	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑦 ∈ V)
16		vex 2752	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
17	16	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑥 ∈ V)
18		eqid 2187	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑆)
19		eqid 2187	. . . . . . . . . . . . 13 ⊢ (.r‘𝑆) = (.r‘𝑆)
20		eqid 2187	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
21		eqid 2187	. . . . . . . . . . . . 13 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
22	18, 19, 20, 21	opprmulg 13304	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Ring ∧ 𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(.r‘(oppr‘𝑆))𝑥) = (𝑥(.r‘𝑆)𝑦))
23	13, 15, 17, 22	syl3anc 1248	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦(.r‘(oppr‘𝑆))𝑥) = (𝑥(.r‘𝑆)𝑦))
24	4	fveq2d 5531	. . . . . . . . . . . 12 ⊢ (𝜑 → (.r‘𝑆) = (.r‘(oppr‘𝑅)))
25	24	oveqd 5905	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥(.r‘𝑆)𝑦) = (𝑥(.r‘(oppr‘𝑅))𝑦))
26		eqid 2187	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
27		eqid 2187	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
28		eqid 2187	. . . . . . . . . . . . 13 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
29	26, 27, 10, 28	opprmulg 13304	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
30	6, 17, 15, 29	syl3anc 1248	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
31	23, 25, 30	3eqtrrd 2225	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦(.r‘𝑅)𝑥) = (𝑦(.r‘(oppr‘𝑆))𝑥))
32	31	eqeq1d 2196	. . . . . . . . 9 ⊢ (𝜑 → ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ↔ (𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅)))
33	32	rexbidv 2488	. . . . . . . 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 2188	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
36		eqidd 2188	. . . . . . . 8 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑅))
37	35, 3, 8, 36	dvdsrd 13325	. . . . . . 7 ⊢ (𝜑 → (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))))
38	10, 26	opprbasg 13308	. . . . . . . . . 10 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
39	8, 38	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
40	4	fveq2d 5531	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(oppr‘𝑅)))
41	20, 18	opprbasg 13308	. . . . . . . . . 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 2226	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(oppr‘𝑆)))
44		eqidd 2188	. . . . . . . 8 ⊢ (𝜑 → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
45	20	opprring 13310	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → (oppr‘𝑆) ∈ Ring)
46	13, 45	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (oppr‘𝑆) ∈ Ring)
47		ringsrg 13282	. . . . . . . . 9 ⊢ ((oppr‘𝑆) ∈ Ring → (oppr‘𝑆) ∈ SRing)
48	46, 47	syl 14	. . . . . . . 8 ⊢ (𝜑 → (oppr‘𝑆) ∈ SRing)
49		eqidd 2188	. . . . . . . 8 ⊢ (𝜑 → (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆)))
50	43, 44, 48, 49	dvdsrd 13325	. . . . . . 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 2188	. . . 4 ⊢ (𝜑 → (Unit‘𝑆) = (Unit‘𝑆))
56		eqid 2187	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
57	10, 56	oppr1g 13313	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘(oppr‘𝑅)))
58	6, 57	syl 14	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) = (1r‘(oppr‘𝑅)))
59	4	fveq2d 5531	. . . . 5 ⊢ (𝜑 → (1r‘𝑆) = (1r‘(oppr‘𝑅)))
60	58, 59	eqtr4d 2223	. . . 4 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑆))
61		eqidd 2188	. . . 4 ⊢ (𝜑 → (oppr‘𝑆) = (oppr‘𝑆))
62		ringsrg 13282	. . . . 5 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
63	13, 62	syl 14	. . . 4 ⊢ (𝜑 → 𝑆 ∈ SRing)
64	55, 60, 5, 61, 44, 63	isunitd 13337	. . 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 2185	1 ⊢ (𝜑 → 𝑈 = (Unit‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1363   ∈ wcel 2158  ∃wrex 2466  Vcvv 2749   class class class wbr 4015  ‘cfv 5228  (class class class)co 5888  Basecbs 12475  ._rcmulr 12551  1_rcur 13196  SRingcsrg 13200  Ringcrg 13233  opp_rcoppr 13300  ∥_rcdsr 13317  Unitcui 13318

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-tpos 6259  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12837  df-grp 12899  df-minusg 12900  df-cmn 13120  df-abl 13121  df-mgp 13163  df-ur 13197  df-srg 13201  df-ring 13235  df-oppr 13301  df-dvdsr 13320  df-unit 13321

This theorem is referenced by: (None)