Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isdmn3 Structured version   Visualization version   GIF version

Theorem isdmn3 38061
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 38042 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2 isdmn3.1 . . . . . 6 𝐺 = (1st𝑅)
3 isdmn3.4 . . . . . 6 𝑍 = (GId‘𝐺)
42, 3isprrngo 38037 . . . . 5 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
5 isdmn3.2 . . . . . . 7 𝐻 = (2nd𝑅)
6 isdmn3.3 . . . . . . 7 𝑋 = ran 𝐺
72, 5, 6ispridlc 38057 . . . . . 6 (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
8 crngorngo 37987 . . . . . . 7 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
98biantrurd 532 . . . . . 6 (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))))
10 3anass 1094 . . . . . . 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 38012 . . . . . . . . . 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 5951 . . . . . . . . . . . . . . 15 ran 𝐺 = ran (1st𝑅)
156, 14eqtri 2763 . . . . . . . . . . . . . 14 𝑋 = ran (1st𝑅)
16 isdmn3.5 . . . . . . . . . . . . . 14 𝑈 = (GId‘𝐻)
1715, 5, 16rngo1cl 37926 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → 𝑈𝑋)
18 eleq2 2828 . . . . . . . . . . . . . 14 ({𝑍} = 𝑋 → (𝑈 ∈ {𝑍} ↔ 𝑈𝑋))
19 elsni 4648 . . . . . . . . . . . . . 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 37927 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
232, 6, 3rngo0cl 37906 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → 𝑍𝑋)
24 en1eqsn 9306 . . . . . . . . . . . . . . . 16 ((𝑍𝑋𝑋 ≈ 1o) → 𝑋 = {𝑍})
2524eqcomd 2741 . . . . . . . . . . . . . . 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 2983 . . . . . . . . 9 (𝑅 ∈ CRingOps → ({𝑍} ≠ 𝑋𝑈𝑍))
32 ovex 7464 . . . . . . . . . . . . 13 (𝑎𝐻𝑏) ∈ V
3332elsn 4646 . . . . . . . . . . . 12 ((𝑎𝐻𝑏) ∈ {𝑍} ↔ (𝑎𝐻𝑏) = 𝑍)
34 velsn 4647 . . . . . . . . . . . . 13 (𝑎 ∈ {𝑍} ↔ 𝑎 = 𝑍)
35 velsn 4647 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑍} ↔ 𝑏 = 𝑍)
3634, 35orbi12i 914 . . . . . . . . . . . 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 3219 . . . . . . . . 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 1094 . . 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 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  {csn 4631   class class class wbr 5148  ran crn 5690  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  1oc1o 8498  cen 8981  GIdcgi 30519  RingOpscrngo 37881  CRingOpsccring 37980  Idlcidl 37994  PrIdlcpridl 37995  PrRingcprrng 38033  Dmncdmn 38034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-en 8985  df-grpo 30522  df-gid 30523  df-ginv 30524  df-ablo 30574  df-ass 37830  df-exid 37832  df-mgmOLD 37836  df-sgrOLD 37848  df-mndo 37854  df-rngo 37882  df-com2 37977  df-crngo 37981  df-idl 37997  df-pridl 37998  df-prrngo 38035  df-dmn 38036  df-igen 38047
This theorem is referenced by:  dmnnzd  38062
  Copyright terms: Public domain W3C validator