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Theorem isdmn3 35220
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 35201 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2 isdmn3.1 . . . . . 6 𝐺 = (1st𝑅)
3 isdmn3.4 . . . . . 6 𝑍 = (GId‘𝐺)
42, 3isprrngo 35196 . . . . 5 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅)))
5 isdmn3.2 . . . . . . 7 𝐻 = (2nd𝑅)
6 isdmn3.3 . . . . . . 7 𝑋 = ran 𝐺
72, 5, 6ispridlc 35216 . . . . . 6 (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ ({𝑍} ∈ (Idl‘𝑅) ∧ {𝑍} ≠ 𝑋 ∧ ∀𝑎𝑋𝑏𝑋 ((𝑎𝐻𝑏) ∈ {𝑍} → (𝑎 ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
8 crngorngo 35146 . . . . . . 7 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
98biantrurd 533 . . . . . 6 (𝑅 ∈ CRingOps → ({𝑍} ∈ (PrIdl‘𝑅) ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))))
10 3anass 1089 . . . . . . 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 35171 . . . . . . . . . 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 5806 . . . . . . . . . . . . . . 15 ran 𝐺 = ran (1st𝑅)
156, 14eqtri 2849 . . . . . . . . . . . . . 14 𝑋 = ran (1st𝑅)
16 isdmn3.5 . . . . . . . . . . . . . 14 𝑈 = (GId‘𝐻)
1715, 5, 16rngo1cl 35085 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → 𝑈𝑋)
18 eleq2 2906 . . . . . . . . . . . . . 14 ({𝑍} = 𝑋 → (𝑈 ∈ {𝑍} ↔ 𝑈𝑋))
19 elsni 4581 . . . . . . . . . . . . . 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 35086 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps → (𝑋 ≈ 1o𝑈 = 𝑍))
232, 6, 3rngo0cl 35065 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps → 𝑍𝑋)
24 en1eqsn 8737 . . . . . . . . . . . . . . . 16 ((𝑍𝑋𝑋 ≈ 1o) → 𝑋 = {𝑍})
2524eqcomd 2832 . . . . . . . . . . . . . . 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 3065 . . . . . . . . 9 (𝑅 ∈ CRingOps → ({𝑍} ≠ 𝑋𝑈𝑍))
32 ovex 7181 . . . . . . . . . . . . 13 (𝑎𝐻𝑏) ∈ V
3332elsn 4579 . . . . . . . . . . . 12 ((𝑎𝐻𝑏) ∈ {𝑍} ↔ (𝑎𝐻𝑏) = 𝑍)
34 velsn 4580 . . . . . . . . . . . . 13 (𝑎 ∈ {𝑍} ↔ 𝑎 = 𝑍)
35 velsn 4580 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑍} ↔ 𝑏 = 𝑍)
3634, 35orbi12i 910 . . . . . . . . . . . 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 3204 . . . . . . . . 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 1089 . . 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 843  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  {csn 4564   class class class wbr 5063  ran crn 5555  cfv 6352  (class class class)co 7148  1st c1st 7678  2nd c2nd 7679  1oc1o 8086  cen 8495  GIdcgi 28181  RingOpscrngo 35040  CRingOpsccring 35139  Idlcidl 35153  PrIdlcpridl 35154  PrRingcprrng 35192  Dmncdmn 35193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-1o 8093  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-grpo 28184  df-gid 28185  df-ginv 28186  df-ablo 28236  df-ass 34989  df-exid 34991  df-mgmOLD 34995  df-sgrOLD 35007  df-mndo 35013  df-rngo 35041  df-com2 35136  df-crngo 35140  df-idl 35156  df-pridl 35157  df-prrngo 35194  df-dmn 35195  df-igen 35206
This theorem is referenced by:  dmnnzd  35221
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