Theorem isdmn3 39148

Description: Obsolete theorem, use isidom3 49581 instead. The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

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 39129	. 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 39124	. . . . 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 39144	. . . . . 6 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
8		crngorngo 39074	. . . . . . 7 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
9	8	biantrurd 542	. . . . . 6 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))))
10		3anass 1111	. . . . . . 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 39099	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
12	8, 11	syl 18	. . . . . . . . 9 ⊢ (𝑅 ∈ CRingOps → {𝑍} ∈ (Idl‘𝑅))
13	12	biantrurd 542	. . . . . . . 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 5923	. . . . . . . . . . . . . . 15 ⊢ ran 𝐺 = ran (1st ‘𝑅)
15	6, 14	eqtri 2786	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran (1st ‘𝑅)
16		isdmn3.5	. . . . . . . . . . . . . 14 ⊢ 𝑈 = (GId‘𝐻)
17	15, 5, 16	rngo1cl 39013	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
18		eleq2 2852	. . . . . . . . . . . . . 14 ⊢ ({𝑍} = 𝑋 → (𝑈 ∈ {𝑍} ↔ 𝑈 ∈ 𝑋))
19		elsni 4600	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
20	18, 19	biimtrrdi 257	. . . . . . . . . . . . 13 ⊢ ({𝑍} = 𝑋 → (𝑈 ∈ 𝑋 → 𝑈 = 𝑍))
21	17, 20	syl5com 32	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → ({𝑍} = 𝑋 → 𝑈 = 𝑍))
22	2, 5, 3, 16, 6	rngoueqz 39014	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
23	2, 6, 3	rngo0cl 38993	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
24		en1eqsn 9277	. . . . . . . . . . . . . . . 16 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {𝑍})
25	24	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → {𝑍} = 𝑋)
26	25	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ 𝑋 → (𝑋 ≈ 1o → {𝑍} = 𝑋))
27	23, 26	syl 18	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o → {𝑍} = 𝑋))
28	22, 27	sylbird 263	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → (𝑈 = 𝑍 → {𝑍} = 𝑋))
29	21, 28	impbid 215	. . . . . . . . . . 11 ⊢ (𝑅 ∈ RingOps → ({𝑍} = 𝑋 ↔ 𝑈 = 𝑍))
30	8, 29	syl 18	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRingOps → ({𝑍} = 𝑋 ↔ 𝑈 = 𝑍))
31	30	necon3bid 3002	. . . . . . . . 9 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ≠ 𝑋 ↔ 𝑈 ≠ 𝑍))
32		ovex 7460	. . . . . . . . . . . . 13 ⊢ (𝑎𝐻𝑏) ∈ V
33	32	elsn 4598	. . . . . . . . . . . 12 ⊢ ((𝑎𝐻𝑏) ∈ {𝑍} ↔ (𝑎𝐻𝑏) = 𝑍)
34		velsn 4599	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ {𝑍} ↔ 𝑎 = 𝑍)
35		velsn 4599	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑍} ↔ 𝑏 = 𝑍)
36	34, 35	orbi12i 928	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}) ↔ (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))
37	33, 36	imbi12i 353	. . . . . . . . . . 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 3229	. . . . . . . . 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 644	. . . . . . . 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 284	. . . . . . 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 286	. . . . . 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 312	. . . . 5 ⊢ (𝑅 ∈ CRingOps → ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)) ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
44	4, 43	bitrid 286	. . . 4 ⊢ (𝑅 ∈ CRingOps → (𝑅 ∈ PrRing ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
45	44	pm5.32i 585	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing) ↔ (𝑅 ∈ CRingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
46		ancom 466	. . 3 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing))
47		3anass 1111	. . 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 306	. 2 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))
49	1, 48	bitri 278	1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146   ≠ wne 2958  ∀wral 3079  {csn 4587   class class class wbr 5105  ran crn 5655  ‘cfv 6546  (class class class)co 7427  1^st c1st 8006  2^nd c2nd 8007  1_oc1o 8480   ≈ cen 8981  GIdcgi 31104  RingOpscrngo 38968  CRingOpsccring 39067  Idlcidl 39081  PrIdlcpridl 39082  PrRingcprrng 39120  Dmncdmn 39121

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-1o 8487  df-en 8985  df-grpo 31107  df-gid 31108  df-ginv 31109  df-ablo 31159  df-ass 38917  df-exid 38919  df-mgmOLD 38923  df-sgrOLD 38935  df-mndo 38941  df-rngo 38969  df-com2 39064  df-crngo 39068  df-idl 39084  df-pridl 39085  df-prrngo 39122  df-dmn 39123  df-igen 39134

This theorem is used by:  dmnnzd  39149