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Theorem drngoi 36036

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 4801	. . . . . 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 2797	. . . . . . . . . . 11 ⊢ (𝑔 = (1^st ‘𝑅) → 𝑔 = 𝐺)
6	5	rneqd 5836	. . . . . . . . . 10 ⊢ (𝑔 = (1^st ‘𝑅) → ran 𝑔 = ran 𝐺)
7		drngi.3	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
8	6, 7	eqtr4di 2797	. . . . . . . . 9 ⊢ (𝑔 = (1^st ‘𝑅) → ran 𝑔 = 𝑋)
9	5	fveq2d 6760	. . . . . . . . . . 11 ⊢ (𝑔 = (1^st ‘𝑅) → (GId‘𝑔) = (GId‘𝐺))
10		drngi.4	. . . . . . . . . . 11 ⊢ 𝑍 = (GId‘𝐺)
11	9, 10	eqtr4di 2797	. . . . . . . . . 10 ⊢ (𝑔 = (1^st ‘𝑅) → (GId‘𝑔) = 𝑍)
12	11	sneqd 4570	. . . . . . . . 9 ⊢ (𝑔 = (1^st ‘𝑅) → {(GId‘𝑔)} = {𝑍})
13	8, 12	difeq12d 4054	. . . . . . . 8 ⊢ (𝑔 = (1^st ‘𝑅) → (ran 𝑔 ∖ {(GId‘𝑔)}) = (𝑋 ∖ {𝑍}))
14	13	sqxpeqd 5612	. . . . . . 7 ⊢ (𝑔 = (1^st ‘𝑅) → ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)})) = ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})))
15	14	reseq2d 5880	. . . . . 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 630	. . . 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 4802	. . . . . . 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 629	. . . . 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 2798	. . . . . . . 8 ⊢ (ℎ = (2^nd ‘𝑅) → 𝐻 = ℎ)
24	23	reseq1d 5879	. . . . . . 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 628	. . . . 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 7875	. . 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 36034	. . 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 5719	. . . . 5 ⊢ Rel DivRingOps
32		1st2nd 7853	. . . . 5 ⊢ ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1^st ‘𝑅), (2^nd ‘𝑅)⟩)
33	31, 32	mpan 686	. . . 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 629	. 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 395 = wceq 1539 ∈ wcel 2108 ∖ cdif 3880 {csn 4558 ⟨cop 4564 {copab 5132 × cxp 5578 ran crn 5581 ↾ cres 5582 Rel wrel 5585 ‘cfv 6418 1^st c1st 7802 2^nd c2nd 7803 GrpOpcgr 28752 GIdcgi 28753 RingOpscrngo 35979 DivRingOpscdrng 36033

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-iota 6376 df-fun 6420 df-fv 6426 df-1st 7804 df-2nd 7805 df-drngo 36034

This theorem is referenced by: dvrunz 36039 fldcrng 36089

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