Theorem isdmn3 38412

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 38393	. 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 38388	. . . . 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 38408	. . . . . 6 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
8		crngorngo 38338	. . . . . . 7 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
9	8	biantrurd 532	. . . . . 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 38363	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
12	8, 11	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRingOps → {𝑍} ∈ (Idl‘𝑅))
13	12	biantrurd 532	. . . . . . . 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 5887	. . . . . . . . . . . . . . 15 ⊢ ran 𝐺 = ran (1st ‘𝑅)
15	6, 14	eqtri 2760	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran (1st ‘𝑅)
16		isdmn3.5	. . . . . . . . . . . . . 14 ⊢ 𝑈 = (GId‘𝐻)
17	15, 5, 16	rngo1cl 38277	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
18		eleq2 2826	. . . . . . . . . . . . . 14 ⊢ ({𝑍} = 𝑋 → (𝑈 ∈ {𝑍} ↔ 𝑈 ∈ 𝑋))
19		elsni 4585	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
20	18, 19	biimtrrdi 254	. . . . . . . . . . . . 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 38278	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
23	2, 6, 3	rngo0cl 38257	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
24		en1eqsn 9179	. . . . . . . . . . . . . . . 16 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {𝑍})
25	24	eqcomd 2743	. . . . . . . . . . . . . . 15 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → {𝑍} = 𝑋)
26	25	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ 𝑋 → (𝑋 ≈ 1o → {𝑍} = 𝑋))
27	23, 26	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o → {𝑍} = 𝑋))
28	22, 27	sylbird 260	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → (𝑈 = 𝑍 → {𝑍} = 𝑋))
29	21, 28	impbid 212	. . . . . . . . . . 11 ⊢ (𝑅 ∈ RingOps → ({𝑍} = 𝑋 ↔ 𝑈 = 𝑍))
30	8, 29	syl 17	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRingOps → ({𝑍} = 𝑋 ↔ 𝑈 = 𝑍))
31	30	necon3bid 2977	. . . . . . . . 9 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ≠ 𝑋 ↔ 𝑈 ≠ 𝑍))
32		ovex 7394	. . . . . . . . . . . . 13 ⊢ (𝑎𝐻𝑏) ∈ V
33	32	elsn 4583	. . . . . . . . . . . 12 ⊢ ((𝑎𝐻𝑏) ∈ {𝑍} ↔ (𝑎𝐻𝑏) = 𝑍)
34		velsn 4584	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ {𝑍} ↔ 𝑎 = 𝑍)
35		velsn 4584	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑍} ↔ 𝑏 = 𝑍)
36	34, 35	orbi12i 915	. . . . . . . . . . . 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 3202	. . . . . . . . 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 633	. . . . . . . 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 281	. . . . . . 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 283	. . . . . 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 309	. . . . 5 ⊢ (𝑅 ∈ CRingOps → ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)) ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
44	4, 43	bitrid 283	. . . 4 ⊢ (𝑅 ∈ CRingOps → (𝑅 ∈ PrRing ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
45	44	pm5.32i 574	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing) ↔ (𝑅 ∈ CRingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
46		ancom 460	. . 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 303	. 2 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))
49	1, 48	bitri 275	1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {csn 4568   class class class wbr 5086  ran crn 5626  ‘cfv 6493  (class class class)co 7361  1^st c1st 7934  2^nd c2nd 7935  1_oc1o 8392   ≈ cen 8884  GIdcgi 30579  RingOpscrngo 38232  CRingOpsccring 38331  Idlcidl 38345  PrIdlcpridl 38346  PrRingcprrng 38384  Dmncdmn 38385

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-1o 8399  df-en 8888  df-grpo 30582  df-gid 30583  df-ginv 30584  df-ablo 30634  df-ass 38181  df-exid 38183  df-mgmOLD 38187  df-sgrOLD 38199  df-mndo 38205  df-rngo 38233  df-com2 38328  df-crngo 38332  df-idl 38348  df-pridl 38349  df-prrngo 38386  df-dmn 38387  df-igen 38398

This theorem is referenced by:  dmnnzd  38413