Theorem isdmn3 36533

Description: The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
isdmn3.1	⊢ 𝐺 = (1st ‘𝑅)
isdmn3.2	⊢ 𝐻 = (2nd ‘𝑅)
isdmn3.3	⊢ 𝑋 = ran 𝐺
isdmn3.4	⊢ 𝑍 = (GId‘𝐺)
isdmn3.5	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
isdmn3	⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))

Distinct variable groups:   𝑅,𝑎,𝑏   𝑍,𝑎,𝑏   𝐻,𝑎,𝑏   𝑋,𝑎,𝑏

Allowed substitution hints:   𝑈(𝑎,𝑏)   𝐺(𝑎,𝑏)

Proof of Theorem isdmn3

Step	Hyp	Ref	Expression
1		isdmn2 36514	. 2 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2		isdmn3.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
3		isdmn3.4	. . . . . 6 ⊢ 𝑍 = (GId‘𝐺)
4	2, 3	isprrngo 36509	. . . . 5 ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
5		isdmn3.2	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
6		isdmn3.3	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
7	2, 5, 6	ispridlc 36529	. . . . . 6 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
8		crngorngo 36459	. . . . . . 7 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
9	8	biantrurd 533	. . . . . 6 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))))
10		3anass 1095	. . . . . . 7 ⊢ (({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ ({𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
11	2, 3	0idl 36484	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
12	8, 11	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRingOps → {𝑍} ∈ (Idl‘𝑅))
13	12	biantrurd 533	. . . . . . . 8 ⊢ (𝑅 ∈ CRingOps → (({𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ ({𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))))))
14	2	rneqi 5892	. . . . . . . . . . . . . . 15 ⊢ ran 𝐺 = ran (1st ‘𝑅)
15	6, 14	eqtri 2764	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran (1st ‘𝑅)
16		isdmn3.5	. . . . . . . . . . . . . 14 ⊢ 𝑈 = (GId‘𝐻)
17	15, 5, 16	rngo1cl 36398	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
18		eleq2 2826	. . . . . . . . . . . . . 14 ⊢ ({𝑍} = 𝑋 → (𝑈 ∈ {𝑍} ↔ 𝑈 ∈ 𝑋))
19		elsni 4603	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
20	18, 19	syl6bir 253	. . . . . . . . . . . . 13 ⊢ ({𝑍} = 𝑋 → (𝑈 ∈ 𝑋 → 𝑈 = 𝑍))
21	17, 20	syl5com 31	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → ({𝑍} = 𝑋 → 𝑈 = 𝑍))
22	2, 5, 3, 16, 6	rngoueqz 36399	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
23	2, 6, 3	rngo0cl 36378	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
24		en1eqsn 9218	. . . . . . . . . . . . . . . 16 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {𝑍})
25	24	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → {𝑍} = 𝑋)
26	25	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ 𝑋 → (𝑋 ≈ 1o → {𝑍} = 𝑋))
27	23, 26	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o → {𝑍} = 𝑋))
28	22, 27	sylbird 259	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → (𝑈 = 𝑍 → {𝑍} = 𝑋))
29	21, 28	impbid 211	. . . . . . . . . . 11 ⊢ (𝑅 ∈ RingOps → ({𝑍} = 𝑋 ↔ 𝑈 = 𝑍))
30	8, 29	syl 17	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRingOps → ({𝑍} = 𝑋 ↔ 𝑈 = 𝑍))
31	30	necon3bid 2988	. . . . . . . . 9 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ≠ 𝑋 ↔ 𝑈 ≠ 𝑍))
32		ovex 7390	. . . . . . . . . . . . 13 ⊢ (𝑎𝐻𝑏) ∈ V
33	32	elsn 4601	. . . . . . . . . . . 12 ⊢ ((𝑎𝐻𝑏) ∈ {𝑍} ↔ (𝑎𝐻𝑏) = 𝑍)
34		velsn 4602	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ {𝑍} ↔ 𝑎 = 𝑍)
35		velsn 4602	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑍} ↔ 𝑏 = 𝑍)
36	34, 35	orbi12i 913	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}) ↔ (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))
37	33, 36	imbi12i 350	. . . . . . . . . . 11 ⊢ (((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))
38	37	a1i 11	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRingOps → (((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))
39	38	2ralbidv 3212	. . . . . . . . 9 ⊢ (𝑅 ∈ CRingOps → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))
40	31, 39	anbi12d 631	. . . . . . . 8 ⊢ (𝑅 ∈ CRingOps → (({𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
41	13, 40	bitr3d 280	. . . . . . 7 ⊢ (𝑅 ∈ CRingOps → (({𝑍} ∈ (Idl‘𝑅) ∧ ({𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))) ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
42	10, 41	bitrid 282	. . . . . 6 ⊢ (𝑅 ∈ CRingOps → (({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
43	7, 9, 42	3bitr3d 308	. . . . 5 ⊢ (𝑅 ∈ CRingOps → ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)) ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
44	4, 43	bitrid 282	. . . 4 ⊢ (𝑅 ∈ CRingOps → (𝑅 ∈ PrRing ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
45	44	pm5.32i 575	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing) ↔ (𝑅 ∈ CRingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
46		ancom 461	. . 3 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing))
47		3anass 1095	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))) ↔ (𝑅 ∈ CRingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
48	45, 46, 47	3bitr4i 302	. 2 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))
49	1, 48	bitri 274	1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {csn 4586   class class class wbr 5105  ran crn 5634  ‘cfv 6496  (class class class)co 7357  1^st c1st 7919  2^nd c2nd 7920  1_oc1o 8405   ≈ cen 8880  GIdcgi 29432  RingOpscrngo 36353  CRingOpsccring 36452  Idlcidl 36466  PrIdlcpridl 36467  PrRingcprrng 36505  Dmncdmn 36506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-1o 8412  df-en 8884  df-grpo 29435  df-gid 29436  df-ginv 29437  df-ablo 29487  df-ass 36302  df-exid 36304  df-mgmOLD 36308  df-sgrOLD 36320  df-mndo 36326  df-rngo 36354  df-com2 36449  df-crngo 36453  df-idl 36469  df-pridl 36470  df-prrngo 36507  df-dmn 36508  df-igen 36519

This theorem is referenced by:  dmnnzd  36534