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Theorem drngoi 37997

Description: The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (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 4825	. . . . . 6 ⊢ (𝑔 = (1^st ‘𝑅) → ⟨𝑔, ℎ⟩ = ⟨(1^st ‘𝑅), ℎ⟩)
2	1	eleq1d 2816	. . . . 5 ⊢ (𝑔 = (1^st ‘𝑅) → (⟨𝑔, ℎ⟩ ∈ RingOps ↔ ⟨(1^st ‘𝑅), ℎ⟩ ∈ RingOps))
3		id 22	. . . . . . . . . . . 12 ⊢ (𝑔 = (1^st ‘𝑅) → 𝑔 = (1^st ‘𝑅))
4		drngi.1	. . . . . . . . . . . 12 ⊢ 𝐺 = (1^st ‘𝑅)
5	3, 4	eqtr4di 2784	. . . . . . . . . . 11 ⊢ (𝑔 = (1^st ‘𝑅) → 𝑔 = 𝐺)
6	5	rneqd 5878	. . . . . . . . . 10 ⊢ (𝑔 = (1^st ‘𝑅) → ran 𝑔 = ran 𝐺)
7		drngi.3	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
8	6, 7	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑔 = (1^st ‘𝑅) → ran 𝑔 = 𝑋)
9	5	fveq2d 6826	. . . . . . . . . . 11 ⊢ (𝑔 = (1^st ‘𝑅) → (GId‘𝑔) = (GId‘𝐺))
10		drngi.4	. . . . . . . . . . 11 ⊢ 𝑍 = (GId‘𝐺)
11	9, 10	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑔 = (1^st ‘𝑅) → (GId‘𝑔) = 𝑍)
12	11	sneqd 4588	. . . . . . . . 9 ⊢ (𝑔 = (1^st ‘𝑅) → {(GId‘𝑔)} = {𝑍})
13	8, 12	difeq12d 4077	. . . . . . . 8 ⊢ (𝑔 = (1^st ‘𝑅) → (ran 𝑔 ∖ {(GId‘𝑔)}) = (𝑋 ∖ {𝑍}))
14	13	sqxpeqd 5648	. . . . . . 7 ⊢ (𝑔 = (1^st ‘𝑅) → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
15	14	reseq2d 5928	. . . . . 6 ⊢ (𝑔 = (1^st ‘𝑅) → (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
16	15	eleq1d 2816	. . . . 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 632	. . . 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 4826	. . . . . . 7 ⊢ (ℎ = (2^nd ‘𝑅) → ⟨(1^st ‘𝑅), ℎ⟩ = ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩)
19	18	eleq1d 2816	. . . . . 6 ⊢ (ℎ = (2^nd ‘𝑅) → (⟨(1^st ‘𝑅), ℎ⟩ ∈ RingOps ↔ ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps))
20	19	anbi1d 631	. . . . 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 22	. . . . . . . . 9 ⊢ (ℎ = (2^nd ‘𝑅) → ℎ = (2^nd ‘𝑅))
23	21, 22	eqtr4id 2785	. . . . . . . 8 ⊢ (ℎ = (2^nd ‘𝑅) → 𝐻 = ℎ)
24	23	reseq1d 5927	. . . . . . 7 ⊢ (ℎ = (2^nd ‘𝑅) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
25	24	eleq1d 2816	. . . . . 6 ⊢ (ℎ = (2^nd ‘𝑅) → ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
26	25	anbi2d 630	. . . . 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 282	. . . 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 7994	. . 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 37995	. . 3 ⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
30	28, 29	eleq2s 2849	. 2 ⊢ (𝑅 ∈ DivRingOps → (⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
31	29	relopabiv 5760	. . . . 5 ⊢ Rel DivRingOps
32		1st2nd 7971	. . . . 5 ⊢ ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩)
33	31, 32	mpan 690	. . . 4 ⊢ (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩)
34	33	eleq1d 2816	. . 3 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ↔ ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps))
35	34	anbi1d 631	. 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 257	1 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∖ cdif 3899 {csn 4576 ⟨cop 4582 {copab 5153 × cxp 5614 ran crn 5617 ↾ cres 5618 Rel wrel 5621 ‘cfv 6481 1^st c1st 7919 2^nd c2nd 7920 GrpOpcgr 30467 GIdcgi 30468 RingOpscrngo 37940 DivRingOpscdrng 37994

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-iota 6437 df-fun 6483 df-fv 6489 df-1st 7921 df-2nd 7922 df-drngo 37995

This theorem is referenced by: dvrunz 38000 fldcrngo 38050

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