Theorem isdrngo1 35226

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 35219	. . . 4 ⊢ DivRingOps = {⟨𝑔, ℎ⟩ ∣ (⟨𝑔, ℎ⟩ ∈ RingOps ∧ (ℎ ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}
2	1	relopabi 5687	. . 3 ⊢ Rel DivRingOps
3		1st2nd 7730	. . 3 ⊢ ((Rel DivRingOps ∧ 𝑅 ∈ DivRingOps) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
4	2, 3	mpan 688	. 2 ⊢ (𝑅 ∈ DivRingOps → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
5		relrngo 35166	. . . 4 ⊢ Rel RingOps
6		1st2nd 7730	. . . 4 ⊢ ((Rel RingOps ∧ 𝑅 ∈ RingOps) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
7	5, 6	mpan 688	. . 3 ⊢ (𝑅 ∈ RingOps → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
8	7	adantr 483	. 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 4800	. . . 4 ⊢ ⟨𝐺, 𝐻⟩ = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩
12	11	eqeq2i 2832	. . 3 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ ↔ 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
13	10	fvexi 6677	. . . . . 6 ⊢ 𝐻 ∈ V
14		isdivrngo 35220	. . . . . 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 4570	. . . . . . . . . 10 ⊢ {𝑍} = {(GId‘𝐺)}
19	16, 18	difeq12i 4095	. . . . . . . . 9 ⊢ (𝑋 ∖ {𝑍}) = (ran 𝐺 ∖ {(GId‘𝐺)})
20	19, 19	xpeq12i 5576	. . . . . . . 8 ⊢ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍})) = ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))
21	20	reseq2i 5843	. . . . . . 7 ⊢ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) = (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)})))
22	21	eleq1i 2901	. . . . . 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 280	. . . 4 ⊢ (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))
25		eleq1 2898	. . . . 5 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ DivRingOps ↔ ⟨𝐺, 𝐻⟩ ∈ DivRingOps))
26		eleq1 2898	. . . . . 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 348	. . . 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 260	. . 3 ⊢ (𝑅 = ⟨𝐺, 𝐻⟩ → (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
30	12, 29	sylbir 237	. 2 ⊢ (𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩ → (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)))
31	4, 8, 30	pm5.21nii 382	1 ⊢ (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1530   ∈ wcel 2107  Vcvv 3493   ∖ cdif 3931  {csn 4559  ⟨cop 4565   × cxp 5546  ran crn 5549   ↾ cres 5550  Rel wrel 5553  ‘cfv 6348  1^st c1st 7679  2^nd c2nd 7680  GrpOpcgr 28258  GIdcgi 28259  RingOpscrngo 35164  DivRingOpscdrng 35218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-1st 7681  df-2nd 7682  df-rngo 35165  df-drngo 35219

This theorem is referenced by:  divrngcl  35227  isdrngo2  35228  divrngpr  35323