Theorem opprunitd 13942

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 2207	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑅))
3		eqidd 2207	. . . . . 6 ⊢ (𝜑 → (∥r‘𝑅) = (∥r‘𝑅))
4		opprunitd.2	. . . . . 6 ⊢ (𝜑 → 𝑆 = (oppr‘𝑅))
5		eqidd 2207	. . . . . 6 ⊢ (𝜑 → (∥r‘𝑆) = (∥r‘𝑆))
6		opprunitd.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		ringsrg 13879	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
8	6, 7	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ SRing)
9	1, 2, 3, 4, 5, 8	isunitd 13938	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘𝑆)(1r‘𝑅))))
10		eqid 2206	. . . . . . . . . . . . . . 15 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
11	10	opprring 13911	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
12	6, 11	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → (oppr‘𝑅) ∈ Ring)
13	4, 12	eqeltrd 2283	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Ring)
14		vex 2776	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
15	14	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑦 ∈ V)
16		vex 2776	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
17	16	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑥 ∈ V)
18		eqid 2206	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑆)
19		eqid 2206	. . . . . . . . . . . . 13 ⊢ (.r‘𝑆) = (.r‘𝑆)
20		eqid 2206	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
21		eqid 2206	. . . . . . . . . . . . 13 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
22	18, 19, 20, 21	opprmulg 13903	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Ring ∧ 𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(.r‘(oppr‘𝑆))𝑥) = (𝑥(.r‘𝑆)𝑦))
23	13, 15, 17, 22	syl3anc 1250	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦(.r‘(oppr‘𝑆))𝑥) = (𝑥(.r‘𝑆)𝑦))
24	4	fveq2d 5592	. . . . . . . . . . . 12 ⊢ (𝜑 → (.r‘𝑆) = (.r‘(oppr‘𝑅)))
25	24	oveqd 5973	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥(.r‘𝑆)𝑦) = (𝑥(.r‘(oppr‘𝑅))𝑦))
26		eqid 2206	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
27		eqid 2206	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
28		eqid 2206	. . . . . . . . . . . . 13 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
29	26, 27, 10, 28	opprmulg 13903	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
30	6, 17, 15, 29	syl3anc 1250	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
31	23, 25, 30	3eqtrrd 2244	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦(.r‘𝑅)𝑥) = (𝑦(.r‘(oppr‘𝑆))𝑥))
32	31	eqeq1d 2215	. . . . . . . . 9 ⊢ (𝜑 → ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ↔ (𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅)))
33	32	rexbidv 2508	. . . . . . . 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 2207	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
36		eqidd 2207	. . . . . . . 8 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑅))
37	35, 3, 8, 36	dvdsrd 13926	. . . . . . 7 ⊢ (𝜑 → (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))))
38	10, 26	opprbasg 13907	. . . . . . . . . 10 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
39	8, 38	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
40	4	fveq2d 5592	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(oppr‘𝑅)))
41	20, 18	opprbasg 13907	. . . . . . . . . 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 2245	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(oppr‘𝑆)))
44		eqidd 2207	. . . . . . . 8 ⊢ (𝜑 → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
45	20	opprring 13911	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → (oppr‘𝑆) ∈ Ring)
46	13, 45	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (oppr‘𝑆) ∈ Ring)
47		ringsrg 13879	. . . . . . . . 9 ⊢ ((oppr‘𝑆) ∈ Ring → (oppr‘𝑆) ∈ SRing)
48	46, 47	syl 14	. . . . . . . 8 ⊢ (𝜑 → (oppr‘𝑆) ∈ SRing)
49		eqidd 2207	. . . . . . . 8 ⊢ (𝜑 → (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆)))
50	43, 44, 48, 49	dvdsrd 13926	. . . . . . 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 2207	. . . 4 ⊢ (𝜑 → (Unit‘𝑆) = (Unit‘𝑆))
56		eqid 2206	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
57	10, 56	oppr1g 13914	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘(oppr‘𝑅)))
58	6, 57	syl 14	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) = (1r‘(oppr‘𝑅)))
59	4	fveq2d 5592	. . . . 5 ⊢ (𝜑 → (1r‘𝑆) = (1r‘(oppr‘𝑅)))
60	58, 59	eqtr4d 2242	. . . 4 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑆))
61		eqidd 2207	. . . 4 ⊢ (𝜑 → (oppr‘𝑆) = (oppr‘𝑆))
62		ringsrg 13879	. . . . 5 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
63	13, 62	syl 14	. . . 4 ⊢ (𝜑 → 𝑆 ∈ SRing)
64	55, 60, 5, 61, 44, 63	isunitd 13938	. . 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 2204	1 ⊢ (𝜑 → 𝑈 = (Unit‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ∃wrex 2486  Vcvv 2773   class class class wbr 4050  ‘cfv 5279  (class class class)co 5956  Basecbs 12902  ._rcmulr 12980  1_rcur 13791  SRingcsrg 13795  Ringcrg 13828  opp_rcoppr 13899  ∥_rcdsr 13918  Unitcui 13919

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-addcom 8040  ax-addass 8042  ax-i2m1 8045  ax-0lt1 8046  ax-0id 8048  ax-rnegex 8049  ax-pre-ltirr 8052  ax-pre-lttrn 8054  ax-pre-ltadd 8056

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-tpos 6343  df-pnf 8124  df-mnf 8125  df-ltxr 8127  df-inn 9052  df-2 9110  df-3 9111  df-ndx 12905  df-slot 12906  df-base 12908  df-sets 12909  df-plusg 12992  df-mulr 12993  df-0g 13160  df-mgm 13258  df-sgrp 13304  df-mnd 13319  df-grp 13405  df-minusg 13406  df-cmn 13692  df-abl 13693  df-mgp 13753  df-ur 13792  df-srg 13796  df-ring 13830  df-oppr 13900  df-dvdsr 13921  df-unit 13922

This theorem is referenced by: (None)