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Theorem isdmn3 35233
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 35214 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2 isdmn3.1 . . . . . 6 𝐺 = (1st𝑅)
3 isdmn3.4 . . . . . 6 𝑍 = (GId‘𝐺)
42, 3isprrngo 35209 . . . . 5 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
5 isdmn3.2 . . . . . . 7 𝐻 = (2nd𝑅)
6 isdmn3.3 . . . . . . 7 𝑋 = ran 𝐺
72, 5, 6ispridlc 35229 . . . . . 6 (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
8 crngorngo 35159 . . . . . . 7 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
98biantrurd 533 . . . . . 6 (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))))
10 3anass 1087 . . . . . . 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 35184 . . . . . . . . . 10 (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))
128, 11syl 17 . . . . . . . . 9 (𝑅 ∈ CRingOps → {𝑍} ∈ (Idl‘𝑅))
1312biantrurd 533 . . . . . . . 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 5800 . . . . . . . . . . . . . . 15 ran 𝐺 = ran (1st𝑅)
156, 14eqtri 2841 . . . . . . . . . . . . . 14 𝑋 = ran (1st𝑅)
16 isdmn3.5 . . . . . . . . . . . . . 14 𝑈 = (GId‘𝐻)
1715, 5, 16rngo1cl 35098 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → 𝑈𝑋)
18 eleq2 2898 . . . . . . . . . . . . . 14 ({𝑍} = 𝑋 → (𝑈 ∈ {𝑍} ↔ 𝑈𝑋))
19 elsni 4574 . . . . . . . . . . . . . 14 (𝑈 ∈ {𝑍} → 𝑈 = 𝑍)
2018, 19syl6bir 255 . . . . . . . . . . . . 13 ({𝑍} = 𝑋 → (𝑈𝑋𝑈 = 𝑍))
2117, 20syl5com 31 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → ({𝑍} = 𝑋𝑈 = 𝑍))
222, 5, 3, 16, 6rngoueqz 35099 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
232, 6, 3rngo0cl 35078 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → 𝑍𝑋)
24 en1eqsn 8736 . . . . . . . . . . . . . . . 16 ((𝑍𝑋𝑋 ≈ 1o) → 𝑋 = {𝑍})
2524eqcomd 2824 . . . . . . . . . . . . . . 15 ((𝑍𝑋𝑋 ≈ 1o) → {𝑍} = 𝑋)
2625ex 413 . . . . . . . . . . . . . 14 (𝑍𝑋 → (𝑋 ≈ 1o → {𝑍} = 𝑋))
2723, 26syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → (𝑋 ≈ 1o → {𝑍} = 𝑋))
2822, 27sylbird 261 . . . . . . . . . . . 12 (𝑅 ∈ RingOps → (𝑈 = 𝑍 → {𝑍} = 𝑋))
2921, 28impbid 213 . . . . . . . . . . 11 (𝑅 ∈ RingOps → ({𝑍} = 𝑋𝑈 = 𝑍))
308, 29syl 17 . . . . . . . . . 10 (𝑅 ∈ CRingOps → ({𝑍} = 𝑋𝑈 = 𝑍))
3130necon3bid 3057 . . . . . . . . 9 (𝑅 ∈ CRingOps → ({𝑍} ≠ 𝑋𝑈𝑍))
32 ovex 7178 . . . . . . . . . . . . 13 (𝑎𝐻𝑏) ∈ V
3332elsn 4572 . . . . . . . . . . . 12 ((𝑎𝐻𝑏) ∈ {𝑍} ↔ (𝑎𝐻𝑏) = 𝑍)
34 velsn 4573 . . . . . . . . . . . . 13 (𝑎 ∈ {𝑍} ↔ 𝑎 = 𝑍)
35 velsn 4573 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑍} ↔ 𝑏 = 𝑍)
3634, 35orbi12i 908 . . . . . . . . . . . 12 ((𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}) ↔ (𝑎 = 𝑍𝑏 = 𝑍))
3733, 36imbi12i 352 . . . . . . . . . . 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 3196 . . . . . . . . 9 (𝑅 ∈ CRingOps → (∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
4031, 39anbi12d 630 . . . . . . . 8 (𝑅 ∈ CRingOps → (({𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4113, 40bitr3d 282 . . . . . . 7 (𝑅 ∈ CRingOps → (({𝑍} ∈ (Idl‘𝑅) ∧ ({𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))) ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4210, 41syl5bb 284 . . . . . 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 310 . . . . 5 (𝑅 ∈ CRingOps → ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)) ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
444, 43syl5bb 284 . . . 4 (𝑅 ∈ CRingOps → (𝑅 ∈ PrRing ↔ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4544pm5.32i 575 . . 3 ((𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing) ↔ (𝑅 ∈ CRingOps ∧ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
46 ancom 461 . . 3 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing))
47 3anass 1087 . . 3 ((𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))) ↔ (𝑅 ∈ CRingOps ∧ (𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍)))))
4845, 46, 473bitr4i 304 . 2 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
491, 48bitri 276 1 (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈𝑍 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍𝑏 = 𝑍))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  {csn 4557   class class class wbr 5057  ran crn 5549  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  1oc1o 8084  cen 8494  GIdcgi 28194  RingOpscrngo 35053  CRingOpsccring 35152  Idlcidl 35166  PrIdlcpridl 35167  PrRingcprrng 35205  Dmncdmn 35206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-1o 8091  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-grpo 28197  df-gid 28198  df-ginv 28199  df-ablo 28249  df-ass 35002  df-exid 35004  df-mgmOLD 35008  df-sgrOLD 35020  df-mndo 35026  df-rngo 35054  df-com2 35149  df-crngo 35153  df-idl 35169  df-pridl 35170  df-prrngo 35207  df-dmn 35208  df-igen 35219
This theorem is referenced by:  dmnnzd  35234
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