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Theorem isdmn3 35877
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 35858 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2 isdmn3.1 . . . . . 6 𝐺 = (1st𝑅)
3 isdmn3.4 . . . . . 6 𝑍 = (GId‘𝐺)
42, 3isprrngo 35853 . . . . 5 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
5 isdmn3.2 . . . . . . 7 𝐻 = (2nd𝑅)
6 isdmn3.3 . . . . . . 7 𝑋 = ran 𝐺
72, 5, 6ispridlc 35873 . . . . . 6 (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
8 crngorngo 35803 . . . . . . 7 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
98biantrurd 536 . . . . . 6 (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))))
10 3anass 1096 . . . . . . 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 35828 . . . . . . . . . 10 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
128, 11syl 17 . . . . . . . . 9 (𝑅 ∈ CRingOps → {𝑍} ∈ (Idl‘𝑅))
1312biantrurd 536 . . . . . . . 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 5780 . . . . . . . . . . . . . . 15 ran 𝐺 = ran (1st𝑅)
156, 14eqtri 2761 . . . . . . . . . . . . . 14 𝑋 = ran (1st𝑅)
16 isdmn3.5 . . . . . . . . . . . . . 14 𝑈 = (GId‘𝐻)
1715, 5, 16rngo1cl 35742 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → 𝑈𝑋)
18 eleq2 2821 . . . . . . . . . . . . . 14 ({𝑍} = 𝑋 → (𝑈 ∈ {𝑍} ↔ 𝑈𝑋))
19 elsni 4533 . . . . . . . . . . . . . 14 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2018, 19syl6bir 257 . . . . . . . . . . . . 13 ({𝑍} = 𝑋 → (𝑈𝑋𝑈 = 𝑍))
2117, 20syl5com 31 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → ({𝑍} = 𝑋𝑈 = 𝑍))
222, 5, 3, 16, 6rngoueqz 35743 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
232, 6, 3rngo0cl 35722 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → 𝑍𝑋)
24 en1eqsn 8827 . . . . . . . . . . . . . . . 16 ((𝑍𝑋𝑋 ≈ 1o) → 𝑋 = {𝑍})
2524eqcomd 2744 . . . . . . . . . . . . . . 15 ((𝑍𝑋𝑋 ≈ 1o) → {𝑍} = 𝑋)
2625ex 416 . . . . . . . . . . . . . 14 (𝑍𝑋 → (𝑋 ≈ 1o → {𝑍} = 𝑋))
2723, 26syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → (𝑋 ≈ 1o → {𝑍} = 𝑋))
2822, 27sylbird 263 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → (𝑈 = 𝑍 → {𝑍} = 𝑋))
2921, 28impbid 215 . . . . . . . . . . 11 (𝑅 ∈ RingOps → ({𝑍} = 𝑋𝑈 = 𝑍))
308, 29syl 17 . . . . . . . . . 10 (𝑅 ∈ CRingOps → ({𝑍} = 𝑋𝑈 = 𝑍))
3130necon3bid 2978 . . . . . . . . 9 (𝑅 ∈ CRingOps → ({𝑍} ≠ 𝑋𝑈𝑍))
32 ovex 7205 . . . . . . . . . . . . 13 (𝑎𝐻𝑏) ∈ V
3332elsn 4531 . . . . . . . . . . . 12 ((𝑎𝐻𝑏) ∈ {𝑍} ↔ (𝑎𝐻𝑏) = 𝑍)
34 velsn 4532 . . . . . . . . . . . . 13 (𝑎 ∈ {𝑍} ↔ 𝑎 = 𝑍)
35 velsn 4532 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑍} ↔ 𝑏 = 𝑍)
3634, 35orbi12i 914 . . . . . . . . . . . 12 ((𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}) ↔ (𝑎 = 𝑍𝑏 = 𝑍))
3733, 36imbi12i 354 . . . . . . . . . . 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 3111 . . . . . . . . 9 (𝑅 ∈ CRingOps → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
4031, 39anbi12d 634 . . . . . . . 8 (𝑅 ∈ CRingOps → (({𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4113, 40bitr3d 284 . . . . . . 7 (𝑅 ∈ CRingOps → (({𝑍} ∈ (Idl‘𝑅) ∧ ({𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))) ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4210, 41syl5bb 286 . . . . . 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 312 . . . . 5 (𝑅 ∈ CRingOps → ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)) ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
444, 43syl5bb 286 . . . 4 (𝑅 ∈ CRingOps → (𝑅 ∈ PrRing ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4544pm5.32i 578 . . 3 ((𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing) ↔ (𝑅 ∈ CRingOps ∧ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
46 ancom 464 . . 3 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing))
47 3anass 1096 . . 3 ((𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))) ↔ (𝑅 ∈ CRingOps ∧ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4845, 46, 473bitr4i 306 . 2 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
491, 48bitri 278 1 (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 846  w3a 1088   = wceq 1542  wcel 2114  wne 2934  wral 3053  {csn 4516   class class class wbr 5030  ran crn 5526  cfv 6339  (class class class)co 7172  1st c1st 7714  2nd c2nd 7715  1oc1o 8126  cen 8554  GIdcgi 28427  RingOpscrngo 35697  CRingOpsccring 35796  Idlcidl 35810  PrIdlcpridl 35811  PrRingcprrng 35849  Dmncdmn 35850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-om 7602  df-1st 7716  df-2nd 7717  df-1o 8133  df-er 8322  df-en 8558  df-dom 8559  df-sdom 8560  df-fin 8561  df-grpo 28430  df-gid 28431  df-ginv 28432  df-ablo 28482  df-ass 35646  df-exid 35648  df-mgmOLD 35652  df-sgrOLD 35664  df-mndo 35670  df-rngo 35698  df-com2 35793  df-crngo 35797  df-idl 35813  df-pridl 35814  df-prrngo 35851  df-dmn 35852  df-igen 35863
This theorem is referenced by:  dmnnzd  35878
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