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Theorem drngoi 38701

Description: Obsolete theorem, use drngring 20897 and drngmgp 20908 instead. The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
drngi.1	⊢ 𝐺 = (1^st ‘𝑅)
drngi.2	⊢ 𝐻 = (2^nd ‘𝑅)
drngi.3	⊢ 𝑋 = ran 𝐺
drngi.4	⊢ 𝑍 = (GId‘𝐺)

**Assertion**
Ref	Expression
drngoi	⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Proof of Theorem drngoi Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 4833	. . . . . 6 ⊢ (𝑔 = (1^st ‘𝑅) → ⟨𝑔, ℎ⟩ = ⟨(1^st ‘𝑅), ℎ⟩)
2	1	eleq1d 2845	. . . . 5 ⊢ (𝑔 = (1^st ‘𝑅) → (⟨𝑔, ℎ⟩ ∈ RingOps ↔ ⟨(1^st ‘𝑅), ℎ⟩ ∈ RingOps))
3		id 23	. . . . . . . . . . . 12 ⊢ (𝑔 = (1^st ‘𝑅) → 𝑔 = (1^st ‘𝑅))
4		drngi.1	. . . . . . . . . . . 12 ⊢ 𝐺 = (1^st ‘𝑅)
5	3, 4	eqtr4di 2813	. . . . . . . . . . 11 ⊢ (𝑔 = (1^st ‘𝑅) → 𝑔 = 𝐺)
6	5	rneqd 5922	. . . . . . . . . 10 ⊢ (𝑔 = (1^st ‘𝑅) → ran 𝑔 = ran 𝐺)
7		drngi.3	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
8	6, 7	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝑔 = (1^st ‘𝑅) → ran 𝑔 = 𝑋)
9	5	fveq2d 6882	. . . . . . . . . . 11 ⊢ (𝑔 = (1^st ‘𝑅) → (GId‘𝑔) = (GId‘𝐺))
10		drngi.4	. . . . . . . . . . 11 ⊢ 𝑍 = (GId‘𝐺)
11	9, 10	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑔 = (1^st ‘𝑅) → (GId‘𝑔) = 𝑍)
12	11	sneqd 4596	. . . . . . . . 9 ⊢ (𝑔 = (1^st ‘𝑅) → {(GId‘𝑔)} = {𝑍})
13	8, 12	difeq12d 4075	. . . . . . . 8 ⊢ (𝑔 = (1^st ‘𝑅) → (ran 𝑔 ∖ {(GId‘𝑔)}) = (𝑋 ∖ {𝑍}))
14	13	sqxpeqd 5687	. . . . . . 7 ⊢ (𝑔 = (1^st ‘𝑅) → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
15	14	reseq2d 5972	. . . . . 6 ⊢ (𝑔 = (1^st ‘𝑅) → (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
16	15	eleq1d 2845	. . . . 5 ⊢ (𝑔 = (1^st ‘𝑅) → ((ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp ↔ (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
17	2, 16	anbi12d 644	. . . 4 ⊢ (𝑔 = (1^st ‘𝑅) → ((⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp) ↔ (⟨(1^st ‘𝑅), ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
18		opeq2 4834	. . . . . . 7 ⊢ (ℎ = (2^nd ‘𝑅) → ⟨(1^st ‘𝑅), ℎ⟩ = ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩)
19	18	eleq1d 2845	. . . . . 6 ⊢ (ℎ = (2^nd ‘𝑅) → (⟨(1^st ‘𝑅), ℎ⟩ ∈ RingOps ↔ ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps))
20	19	anbi1d 643	. . . . 5 ⊢ (ℎ = (2^nd ‘𝑅) → ((⟨(1^st ‘𝑅), ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps ∧ (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
21		drngi.2	. . . . . . . . 9 ⊢ 𝐻 = (2^nd ‘𝑅)
22		id 23	. . . . . . . . 9 ⊢ (ℎ = (2^nd ‘𝑅) → ℎ = (2^nd ‘𝑅))
23	21, 22	eqtr4id 2814	. . . . . . . 8 ⊢ (ℎ = (2^nd ‘𝑅) → 𝐻 = ℎ)
24	23	reseq1d 5971	. . . . . . 7 ⊢ (ℎ = (2^nd ‘𝑅) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
25	24	eleq1d 2845	. . . . . 6 ⊢ (ℎ = (2^nd ‘𝑅) → ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
26	25	anbi2d 642	. . . . 5 ⊢ (ℎ = (2^nd ‘𝑅) → ((⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps ∧ (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
27	20, 26	bitr4d 285	. . . 4 ⊢ (ℎ = (2^nd ‘𝑅) → ((⟨(1^st ‘𝑅), ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
28	17, 27	elopabi 8059	. . 3 ⊢ (𝑅 ∈ {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)} → (⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
29		df-drngo 38699	. . 3 ⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
30	28, 29	eleq2s 2878	. 2 ⊢ (𝑅 ∈ DivRingOps → (⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
31	29	relopabiv 5801	. . . . 5 ⊢ Rel DivRingOps
32		1st2nd 8036	. . . . 5 ⊢ ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩)
33	31, 32	mpan 703	. . . 4 ⊢ (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩)
34	33	eleq1d 2845	. . 3 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ↔ ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps))
35	34	anbi1d 643	. 2 ⊢ (𝑅 ∈ DivRingOps → ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
36	30, 35	mpbird 260	1 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∖ cdif 3896 {csn 4584 ⟨cop 4590 {copab 5167 × cxp 5653 ran crn 5656 ↾ cres 5657 Rel wrel 5660 ‘cfv 6533 1^st c1st 7984 2^nd c2nd 7985 GrpOpcgr 30970 GIdcgi 30971 RingOpscrngo 38644 DivRingOpscdrng 38698

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7736

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-iota 6489 df-fun 6535 df-fv 6541 df-1st 7986 df-2nd 7987 df-drngo 38699

This theorem is used by: dvrunz 38704 fldcrngo 38754

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