Theorem isdmn3 34414

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 34395	. 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 34390	. . . . 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 34410	. . . . . 6 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
8		crngorngo 34340	. . . . . . 7 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
9	8	biantrurd 530	. . . . . 6 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))))
10		3anass 1122	. . . . . . 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 34365	. . . . . . . . . 10 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
12	8, 11	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ CRingOps → {𝑍} ∈ (Idl‘𝑅))
13	12	biantrurd 530	. . . . . . . 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 5583	. . . . . . . . . . . . . . 15 ⊢ ran 𝐺 = ran (1st ‘𝑅)
15	6, 14	eqtri 2848	. . . . . . . . . . . . . 14 ⊢ 𝑋 = ran (1st ‘𝑅)
16		isdmn3.5	. . . . . . . . . . . . . 14 ⊢ 𝑈 = (GId‘𝐻)
17	15, 5, 16	rngo1cl 34279	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
18		eleq2 2894	. . . . . . . . . . . . . 14 ⊢ ({𝑍} = 𝑋 → (𝑈 ∈ {𝑍} ↔ 𝑈 ∈ 𝑋))
19		elsni 4413	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
20	18, 19	syl6bir 246	. . . . . . . . . . . . 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 34280	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍))
23	2, 6, 3	rngo0cl 34259	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ RingOps → 𝑍 ∈ 𝑋)
24		en1eqsn 8458	. . . . . . . . . . . . . . . 16 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → 𝑋 = {𝑍})
25	24	eqcomd 2830	. . . . . . . . . . . . . . 15 ⊢ ((𝑍 ∈ 𝑋 ∧ 𝑋 ≈ 1o) → {𝑍} = 𝑋)
26	25	ex 403	. . . . . . . . . . . . . 14 ⊢ (𝑍 ∈ 𝑋 → (𝑋 ≈ 1o → {𝑍} = 𝑋))
27	23, 26	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o → {𝑍} = 𝑋))
28	22, 27	sylbird 252	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ RingOps → (𝑈 = 𝑍 → {𝑍} = 𝑋))
29	21, 28	impbid 204	. . . . . . . . . . 11 ⊢ (𝑅 ∈ RingOps → ({𝑍} = 𝑋 ↔ 𝑈 = 𝑍))
30	8, 29	syl 17	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRingOps → ({𝑍} = 𝑋 ↔ 𝑈 = 𝑍))
31	30	necon3bid 3042	. . . . . . . . 9 ⊢ (𝑅 ∈ CRingOps → ({𝑍} ≠ 𝑋 ↔ 𝑈 ≠ 𝑍))
32		ovex 6936	. . . . . . . . . . . . 13 ⊢ (𝑎𝐻𝑏) ∈ V
33	32	elsn 4411	. . . . . . . . . . . 12 ⊢ ((𝑎𝐻𝑏) ∈ {𝑍} ↔ (𝑎𝐻𝑏) = 𝑍)
34		velsn 4412	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ {𝑍} ↔ 𝑎 = 𝑍)
35		velsn 4412	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑍} ↔ 𝑏 = 𝑍)
36	34, 35	orbi12i 945	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}) ↔ (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))
37	33, 36	imbi12i 342	. . . . . . . . . . 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 3197	. . . . . . . . 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 626	. . . . . . . 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 273	. . . . . . 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	syl5bb 275	. . . . . 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 301	. . . . 5 ⊢ (𝑅 ∈ CRingOps → ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)) ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
44	4, 43	syl5bb 275	. . . 4 ⊢ (𝑅 ∈ CRingOps → (𝑅 ∈ PrRing ↔ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
45	44	pm5.32i 572	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing) ↔ (𝑅 ∈ CRingOps ∧ (𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))))
46		ancom 454	. . 3 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing))
47		3anass 1122	. . 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 295	. 2 ⊢ ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))
49	1, 48	bitri 267	1 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 880   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166   ≠ wne 2998  ∀wral 3116  {csn 4396   class class class wbr 4872  ran crn 5342  ‘cfv 6122  (class class class)co 6904  1^st c1st 7425  2^nd c2nd 7426  1_oc1o 7818   ≈ cen 8218  GIdcgi 27899  RingOpscrngo 34234  CRingOpsccring 34333  Idlcidl 34347  PrIdlcpridl 34348  PrRingcprrng 34386  Dmncdmn 34387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-ord 5965  df-on 5966  df-lim 5967  df-suc 5968  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-om 7326  df-1st 7427  df-2nd 7428  df-1o 7825  df-er 8008  df-en 8222  df-dom 8223  df-sdom 8224  df-fin 8225  df-grpo 27902  df-gid 27903  df-ginv 27904  df-ablo 27954  df-ass 34183  df-exid 34185  df-mgmOLD 34189  df-sgrOLD 34201  df-mndo 34207  df-rngo 34235  df-com2 34330  df-crngo 34334  df-idl 34350  df-pridl 34351  df-prrngo 34388  df-dmn 34389  df-igen 34400

This theorem is referenced by:  dmnnzd  34415