Theorem isdrngo1 36762

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 36755	. . . 4 ⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
2	1	relopabiv 5818	. . 3 ⊢ Rel DivRingOps
3		1st2nd 8020	. . 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 36702	. . . 4 ⊢ Rel RingOps
6		1st2nd 8020	. . . 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 482	. 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 4877	. . . 4 ⊢ ⟨𝐺, 𝐻⟩ = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩
12	11	eqeq2i 2746	. . 3 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ ↔ 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
13	10	fvexi 6902	. . . . . 6 ⊢ 𝐻 ∈ V
14		isdivrngo 36756	. . . . . 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 4638	. . . . . . . . . 10 ⊢ {𝑍} = {(GId‘𝐺)}
19	16, 18	difeq12i 4119	. . . . . . . . 9 ⊢ (𝑋 ∖ {𝑍}) = (ran 𝐺 ∖ {(GId‘𝐺)})
20	19, 19	xpeq12i 5703	. . . . . . . 8 ⊢ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) = ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))
21	20	reseq2i 5976	. . . . . . 7 ⊢ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)})))
22	21	eleq1i 2825	. . . . . 6 ⊢ ((𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp ↔ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)
23	22	anbi2i 624	. . . . 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 2822	. . . . 5 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ DivRingOps ↔ ⟨𝐺, 𝐻⟩ ∈ DivRingOps))
26		eleq1 2822	. . . . . 6 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ RingOps ↔ ⟨𝐺, 𝐻⟩ ∈ RingOps))
27	26	anbi1d 631	. . . . 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 346	. . . 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 234	. 2 ⊢ (𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩ → (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
31	4, 8, 30	pm5.21nii 380	1 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∖ cdif 3944  {csn 4627  ⟨cop 4633   × cxp 5673  ran crn 5676   ↾ cres 5677  Rel wrel 5680  ‘cfv 6540  1^st c1st 7968  2^nd c2nd 7969  GrpOpcgr 29720  GIdcgi 29721  RingOpscrngo 36700  DivRingOpscdrng 36754

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7407  df-1st 7970  df-2nd 7971  df-rngo 36701  df-drngo 36755

This theorem is referenced by:  divrngcl  36763  isdrngo2  36764  divrngpr  36859