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Theorem drngoi 36109

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 4804	. . . . . 6 ⊢ (𝑔 = (1^st ‘𝑅) → ⟨𝑔, ℎ⟩ = ⟨(1^st ‘𝑅), ℎ⟩)
2	1	eleq1d 2823	. . . . 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 2796	. . . . . . . . . . 11 ⊢ (𝑔 = (1^st ‘𝑅) → 𝑔 = 𝐺)
6	5	rneqd 5847	. . . . . . . . . 10 ⊢ (𝑔 = (1^st ‘𝑅) → ran 𝑔 = ran 𝐺)
7		drngi.3	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
8	6, 7	eqtr4di 2796	. . . . . . . . 9 ⊢ (𝑔 = (1^st ‘𝑅) → ran 𝑔 = 𝑋)
9	5	fveq2d 6778	. . . . . . . . . . 11 ⊢ (𝑔 = (1^st ‘𝑅) → (GId‘𝑔) = (GId‘𝐺))
10		drngi.4	. . . . . . . . . . 11 ⊢ 𝑍 = (GId‘𝐺)
11	9, 10	eqtr4di 2796	. . . . . . . . . 10 ⊢ (𝑔 = (1^st ‘𝑅) → (GId‘𝑔) = 𝑍)
12	11	sneqd 4573	. . . . . . . . 9 ⊢ (𝑔 = (1^st ‘𝑅) → {(GId‘𝑔)} = {𝑍})
13	8, 12	difeq12d 4058	. . . . . . . 8 ⊢ (𝑔 = (1^st ‘𝑅) → (ran 𝑔 ∖ {(GId‘𝑔)}) = (𝑋 ∖ {𝑍}))
14	13	sqxpeqd 5621	. . . . . . 7 ⊢ (𝑔 = (1^st ‘𝑅) → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
15	14	reseq2d 5891	. . . . . 6 ⊢ (𝑔 = (1^st ‘𝑅) → (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) = (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
16	15	eleq1d 2823	. . . . 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 631	. . . 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 4805	. . . . . . 7 ⊢ (ℎ = (2^nd ‘𝑅) → ⟨(1^st ‘𝑅), ℎ⟩ = ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩)
19	18	eleq1d 2823	. . . . . 6 ⊢ (ℎ = (2^nd ‘𝑅) → (⟨(1^st ‘𝑅), ℎ⟩ ∈ RingOps ↔ ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps))
20	19	anbi1d 630	. . . . 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 2797	. . . . . . . 8 ⊢ (ℎ = (2^nd ‘𝑅) → 𝐻 = ℎ)
24	23	reseq1d 5890	. . . . . . 7 ⊢ (ℎ = (2^nd ‘𝑅) → (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))))
25	24	eleq1d 2823	. . . . . 6 ⊢ (ℎ = (2^nd ‘𝑅) → ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ (ℎ ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
26	25	anbi2d 629	. . . . 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 281	. . . 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 7902	. . 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 36107	. . 3 ⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
30	28, 29	eleq2s 2857	. 2 ⊢ (𝑅 ∈ DivRingOps → (⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
31	29	relopabiv 5730	. . . . 5 ⊢ Rel DivRingOps
32		1st2nd 7880	. . . . 5 ⊢ ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩)
33	31, 32	mpan 687	. . . 4 ⊢ (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩)
34	33	eleq1d 2823	. . 3 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ↔ ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩ ∈ RingOps))
35	34	anbi1d 630	. 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 256	1 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∖ cdif 3884 {csn 4561 ⟨cop 4567 {copab 5136 × cxp 5587 ran crn 5590 ↾ cres 5591 Rel wrel 5594 ‘cfv 6433 1^st c1st 7829 2^nd c2nd 7830 GrpOpcgr 28851 GIdcgi 28852 RingOpscrngo 36052 DivRingOpscdrng 36106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-iota 6391 df-fun 6435 df-fv 6441 df-1st 7831 df-2nd 7832 df-drngo 36107

This theorem is referenced by: dvrunz 36112 fldcrng 36162

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