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Theorem isdmn3 38081
Description: The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
isdmn3.1 𝐺 = (1st𝑅)
isdmn3.2 𝐻 = (2nd𝑅)
isdmn3.3 𝑋 = ran 𝐺
isdmn3.4 𝑍 = (GId‘𝐺)
isdmn3.5 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
isdmn3 (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 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
StepHypRef Expression
1 isdmn2 38062 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2 isdmn3.1 . . . . . 6 𝐺 = (1st𝑅)
3 isdmn3.4 . . . . . 6 𝑍 = (GId‘𝐺)
42, 3isprrngo 38057 . . . . 5 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
5 isdmn3.2 . . . . . . 7 𝐻 = (2nd𝑅)
6 isdmn3.3 . . . . . . 7 𝑋 = ran 𝐺
72, 5, 6ispridlc 38077 . . . . . 6 (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
8 crngorngo 38007 . . . . . . 7 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
98biantrurd 532 . . . . . 6 (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))))
10 3anass 1095 . . . . . . 7 (({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ ({𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
112, 30idl 38032 . . . . . . . . . 10 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
128, 11syl 17 . . . . . . . . 9 (𝑅 ∈ CRingOps → {𝑍} ∈ (Idl‘𝑅))
1312biantrurd 532 . . . . . . . 8 (𝑅 ∈ CRingOps → (({𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ ({𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))))))
142rneqi 5948 . . . . . . . . . . . . . . 15 ran 𝐺 = ran (1st𝑅)
156, 14eqtri 2765 . . . . . . . . . . . . . 14 𝑋 = ran (1st𝑅)
16 isdmn3.5 . . . . . . . . . . . . . 14 𝑈 = (GId‘𝐻)
1715, 5, 16rngo1cl 37946 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → 𝑈𝑋)
18 eleq2 2830 . . . . . . . . . . . . . 14 ({𝑍} = 𝑋 → (𝑈 ∈ {𝑍} ↔ 𝑈𝑋))
19 elsni 4643 . . . . . . . . . . . . . 14 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2018, 19biimtrrdi 254 . . . . . . . . . . . . 13 ({𝑍} = 𝑋 → (𝑈𝑋𝑈 = 𝑍))
2117, 20syl5com 31 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → ({𝑍} = 𝑋𝑈 = 𝑍))
222, 5, 3, 16, 6rngoueqz 37947 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
232, 6, 3rngo0cl 37926 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → 𝑍𝑋)
24 en1eqsn 9308 . . . . . . . . . . . . . . . 16 ((𝑍𝑋𝑋 ≈ 1o) → 𝑋 = {𝑍})
2524eqcomd 2743 . . . . . . . . . . . . . . 15 ((𝑍𝑋𝑋 ≈ 1o) → {𝑍} = 𝑋)
2625ex 412 . . . . . . . . . . . . . 14 (𝑍𝑋 → (𝑋 ≈ 1o → {𝑍} = 𝑋))
2723, 26syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → (𝑋 ≈ 1o → {𝑍} = 𝑋))
2822, 27sylbird 260 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → (𝑈 = 𝑍 → {𝑍} = 𝑋))
2921, 28impbid 212 . . . . . . . . . . 11 (𝑅 ∈ RingOps → ({𝑍} = 𝑋𝑈 = 𝑍))
308, 29syl 17 . . . . . . . . . 10 (𝑅 ∈ CRingOps → ({𝑍} = 𝑋𝑈 = 𝑍))
3130necon3bid 2985 . . . . . . . . 9 (𝑅 ∈ CRingOps → ({𝑍} ≠ 𝑋𝑈𝑍))
32 ovex 7464 . . . . . . . . . . . . 13 (𝑎𝐻𝑏) ∈ V
3332elsn 4641 . . . . . . . . . . . 12 ((𝑎𝐻𝑏) ∈ {𝑍} ↔ (𝑎𝐻𝑏) = 𝑍)
34 velsn 4642 . . . . . . . . . . . . 13 (𝑎 ∈ {𝑍} ↔ 𝑎 = 𝑍)
35 velsn 4642 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑍} ↔ 𝑏 = 𝑍)
3634, 35orbi12i 915 . . . . . . . . . . . 12 ((𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}) ↔ (𝑎 = 𝑍𝑏 = 𝑍))
3733, 36imbi12i 350 . . . . . . . . . . 11 (((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))
3837a1i 11 . . . . . . . . . 10 (𝑅 ∈ CRingOps → (((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
39382ralbidv 3221 . . . . . . . . 9 (𝑅 ∈ CRingOps → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
4031, 39anbi12d 632 . . . . . . . 8 (𝑅 ∈ CRingOps → (({𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4113, 40bitr3d 281 . . . . . . 7 (𝑅 ∈ CRingOps → (({𝑍} ∈ (Idl‘𝑅) ∧ ({𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))) ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4210, 41bitrid 283 . . . . . 6 (𝑅 ∈ CRingOps → (({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
437, 9, 423bitr3d 309 . . . . 5 (𝑅 ∈ CRingOps → ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)) ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
444, 43bitrid 283 . . . 4 (𝑅 ∈ CRingOps → (𝑅 ∈ PrRing ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4544pm5.32i 574 . . 3 ((𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing) ↔ (𝑅 ∈ CRingOps ∧ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
46 ancom 460 . . 3 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing))
47 3anass 1095 . . 3 ((𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))) ↔ (𝑅 ∈ CRingOps ∧ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4845, 46, 473bitr4i 303 . 2 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
491, 48bitri 275 1 (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  {csn 4626   class class class wbr 5143  ran crn 5686  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  1oc1o 8499  cen 8982  GIdcgi 30509  RingOpscrngo 37901  CRingOpsccring 38000  Idlcidl 38014  PrIdlcpridl 38015  PrRingcprrng 38053  Dmncdmn 38054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-en 8986  df-grpo 30512  df-gid 30513  df-ginv 30514  df-ablo 30564  df-ass 37850  df-exid 37852  df-mgmOLD 37856  df-sgrOLD 37868  df-mndo 37874  df-rngo 37902  df-com2 37997  df-crngo 38001  df-idl 38017  df-pridl 38018  df-prrngo 38055  df-dmn 38056  df-igen 38067
This theorem is referenced by:  dmnnzd  38082
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