Theorem isdrngo1 37916

Description: The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)

Hypotheses

Ref	Expression
isdivrng1.1	⊢ 𝐺 = (1st ‘𝑅)
isdivrng1.2	⊢ 𝐻 = (2nd ‘𝑅)
isdivrng1.3	⊢ 𝑍 = (GId‘𝐺)
isdivrng1.4	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
isdrngo1	⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Proof of Theorem isdrngo1

Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-drngo 37909	. . . 4 ⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
2	1	relopabiv 5844	. . 3 ⊢ Rel DivRingOps
3		1st2nd 8080	. . 3 ⊢ ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
4	2, 3	mpan 689	. 2 ⊢ (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
5		relrngo 37856	. . . 4 ⊢ Rel RingOps
6		1st2nd 8080	. . . 4 ⊢ ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
7	5, 6	mpan 689	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
8	7	adantr 480	. 2 ⊢ ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
9		isdivrng1.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
10		isdivrng1.2	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
11	9, 10	opeq12i 4902	. . . 4 ⊢ ⟨𝐺, 𝐻⟩ = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩
12	11	eqeq2i 2753	. . 3 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ ↔ 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
13	10	fvexi 6934	. . . . . 6 ⊢ 𝐻 ∈ V
14		isdivrngo 37910	. . . . . 6 ⊢ (𝐻 ∈ V → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))
15	13, 14	ax-mp 5	. . . . 5 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
16		isdivrng1.4	. . . . . . . . . 10 ⊢ 𝑋 = ran 𝐺
17		isdivrng1.3	. . . . . . . . . . 11 ⊢ 𝑍 = (GId‘𝐺)
18	17	sneqi 4659	. . . . . . . . . 10 ⊢ {𝑍} = {(GId‘𝐺)}
19	16, 18	difeq12i 4147	. . . . . . . . 9 ⊢ (𝑋 ∖ {𝑍}) = (ran 𝐺 ∖ {(GId‘𝐺)})
20	19, 19	xpeq12i 5728	. . . . . . . 8 ⊢ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) = ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))
21	20	reseq2i 6006	. . . . . . 7 ⊢ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)})))
22	21	eleq1i 2835	. . . . . 6 ⊢ ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)
23	22	anbi2i 622	. . . . 5 ⊢ ((⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp))
24	15, 23	bitr4i 278	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
25		eleq1 2832	. . . . 5 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ DivRingOps ↔ ⟨𝐺, 𝐻⟩ ∈ DivRingOps))
26		eleq1 2832	. . . . . 6 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps))
27	26	anbi1d 630	. . . . 5 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ → ((𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp) ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
28	25, 27	bibi12d 345	. . . 4 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ → ((𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)) ↔ (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))))
29	24, 28	mpbiri 258	. . 3 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
30	12, 29	sylbir 235	. 2 ⊢ (𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩ → (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
31	4, 8, 30	pm5.21nii 378	1 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973  {csn 4648  ⟨cop 4654   × cxp 5698  ran crn 5701   ↾ cres 5702  Rel wrel 5705  ‘cfv 6573  1^st c1st 8028  2^nd c2nd 8029  GrpOpcgr 30521  GIdcgi 30522  RingOpscrngo 37854  DivRingOpscdrng 37908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-1st 8030  df-2nd 8031  df-rngo 37855  df-drngo 37909

This theorem is referenced by:  divrngcl  37917  isdrngo2  37918  divrngpr  38013