Theorem opprunitd 14526

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 2239	. . . . . 6 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑅))
3		eqidd 2239	. . . . . 6 ⊢ (𝜑 → (∥r‘𝑅) = (∥r‘𝑅))
4		opprunitd.2	. . . . . 6 ⊢ (𝜑 → 𝑆 = (oppr‘𝑅))
5		eqidd 2239	. . . . . 6 ⊢ (𝜑 → (∥r‘𝑆) = (∥r‘𝑆))
6		opprunitd.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
7		ringsrg 14459	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing)
8	6, 7	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ SRing)
9	1, 2, 3, 4, 5, 8	isunitd 14522	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↔ (𝑥(∥r‘𝑅)(1r‘𝑅) ∧ 𝑥(∥r‘𝑆)(1r‘𝑅))))
10		eqid 2238	. . . . . . . . . . . . . . 15 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
11	10	opprring 14493	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring)
12	6, 11	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → (oppr‘𝑅) ∈ Ring)
13	4, 12	eqeltrd 2315	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Ring)
14		vex 2824	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
15	14	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑦 ∈ V)
16		vex 2824	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
17	16	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑥 ∈ V)
18		eqid 2238	. . . . . . . . . . . . 13 ⊢ (Base‘𝑆) = (Base‘𝑆)
19		eqid 2238	. . . . . . . . . . . . 13 ⊢ (.r‘𝑆) = (.r‘𝑆)
20		eqid 2238	. . . . . . . . . . . . 13 ⊢ (oppr‘𝑆) = (oppr‘𝑆)
21		eqid 2238	. . . . . . . . . . . . 13 ⊢ (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆))
22	18, 19, 20, 21	opprmulg 14485	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Ring ∧ 𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦(.r‘(oppr‘𝑆))𝑥) = (𝑥(.r‘𝑆)𝑦))
23	13, 15, 17, 22	syl3anc 1278	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑦(.r‘(oppr‘𝑆))𝑥) = (𝑥(.r‘𝑆)𝑦))
24	4	fveq2d 5702	. . . . . . . . . . . 12 ⊢ (𝜑 → (.r‘𝑆) = (.r‘(oppr‘𝑅)))
25	24	oveqd 6105	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥(.r‘𝑆)𝑦) = (𝑥(.r‘(oppr‘𝑅))𝑦))
26		eqid 2238	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
27		eqid 2238	. . . . . . . . . . . . 13 ⊢ (.r‘𝑅) = (.r‘𝑅)
28		eqid 2238	. . . . . . . . . . . . 13 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
29	26, 27, 10, 28	opprmulg 14485	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
30	6, 17, 15, 29	syl3anc 1278	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥(.r‘(oppr‘𝑅))𝑦) = (𝑦(.r‘𝑅)𝑥))
31	23, 25, 30	3eqtrrd 2276	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦(.r‘𝑅)𝑥) = (𝑦(.r‘(oppr‘𝑆))𝑥))
32	31	eqeq1d 2247	. . . . . . . . 9 ⊢ (𝜑 → ((𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ↔ (𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅)))
33	32	rexbidv 2551	. . . . . . . 8 ⊢ (𝜑 → (∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅) ↔ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘(oppr‘𝑆))𝑥) = (1r‘𝑅)))
34	33	anbi2d 468	. . . . . . 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 2239	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑅))
36		eqidd 2239	. . . . . . . 8 ⊢ (𝜑 → (.r‘𝑅) = (.r‘𝑅))
37	35, 3, 8, 36	dvdsrd 14510	. . . . . . 7 ⊢ (𝜑 → (𝑥(∥r‘𝑅)(1r‘𝑅) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∃𝑦 ∈ (Base‘𝑅)(𝑦(.r‘𝑅)𝑥) = (1r‘𝑅))))
38	10, 26	opprbasg 14489	. . . . . . . . . 10 ⊢ (𝑅 ∈ SRing → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
39	8, 38	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(oppr‘𝑅)))
40	4	fveq2d 5702	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑆) = (Base‘(oppr‘𝑅)))
41	20, 18	opprbasg 14489	. . . . . . . . . 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 2277	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(oppr‘𝑆)))
44		eqidd 2239	. . . . . . . 8 ⊢ (𝜑 → (∥r‘(oppr‘𝑆)) = (∥r‘(oppr‘𝑆)))
45	20	opprring 14493	. . . . . . . . . 10 ⊢ (𝑆 ∈ Ring → (oppr‘𝑆) ∈ Ring)
46	13, 45	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (oppr‘𝑆) ∈ Ring)
47		ringsrg 14459	. . . . . . . . 9 ⊢ ((oppr‘𝑆) ∈ Ring → (oppr‘𝑆) ∈ SRing)
48	46, 47	syl 14	. . . . . . . 8 ⊢ (𝜑 → (oppr‘𝑆) ∈ SRing)
49		eqidd 2239	. . . . . . . 8 ⊢ (𝜑 → (.r‘(oppr‘𝑆)) = (.r‘(oppr‘𝑆)))
50	43, 44, 48, 49	dvdsrd 14510	. . . . . . 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 469	. . . . 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 2239	. . . 4 ⊢ (𝜑 → (Unit‘𝑆) = (Unit‘𝑆))
56		eqid 2238	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
57	10, 56	oppr1g 14497	. . . . . 6 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘(oppr‘𝑅)))
58	6, 57	syl 14	. . . . 5 ⊢ (𝜑 → (1r‘𝑅) = (1r‘(oppr‘𝑅)))
59	4	fveq2d 5702	. . . . 5 ⊢ (𝜑 → (1r‘𝑆) = (1r‘(oppr‘𝑅)))
60	58, 59	eqtr4d 2274	. . . 4 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝑆))
61		eqidd 2239	. . . 4 ⊢ (𝜑 → (oppr‘𝑆) = (oppr‘𝑆))
62		ringsrg 14459	. . . . 5 ⊢ (𝑆 ∈ Ring → 𝑆 ∈ SRing)
63	13, 62	syl 14	. . . 4 ⊢ (𝜑 → 𝑆 ∈ SRing)
64	55, 60, 5, 61, 44, 63	isunitd 14522	. . 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 2236	1 ⊢ (𝜑 → 𝑈 = (Unit‘𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821   class class class wbr 4131  ‘cfv 5380  (class class class)co 6088  Basecbs 13410  ._rcmulr 13491  1_rcur 14369  SRingcsrg 14374  Ringcrg 14407  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-oppr 14482  df-dvdsr 14504  df-unit 14505

This theorem is used by:  opprlring  14613  opprdrng  14729