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Theorem isdmn3 36562
Description: The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
isdmn3.1 𝐺 = (1st β€˜π‘…)
isdmn3.2 𝐻 = (2nd β€˜π‘…)
isdmn3.3 𝑋 = ran 𝐺
isdmn3.4 𝑍 = (GIdβ€˜πΊ)
isdmn3.5 π‘ˆ = (GIdβ€˜π»)
Assertion
Ref Expression
isdmn3 (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍))))
Distinct variable groups:   𝑅,π‘Ž,𝑏   𝑍,π‘Ž,𝑏   𝐻,π‘Ž,𝑏   𝑋,π‘Ž,𝑏
Allowed substitution hints:   π‘ˆ(π‘Ž,𝑏)   𝐺(π‘Ž,𝑏)

Proof of Theorem isdmn3
StepHypRef Expression
1 isdmn2 36543 . 2 (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
2 isdmn3.1 . . . . . 6 𝐺 = (1st β€˜π‘…)
3 isdmn3.4 . . . . . 6 𝑍 = (GIdβ€˜πΊ)
42, 3isprrngo 36538 . . . . 5 (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdlβ€˜π‘…)))
5 isdmn3.2 . . . . . . 7 𝐻 = (2nd β€˜π‘…)
6 isdmn3.3 . . . . . . 7 𝑋 = ran 𝐺
72, 5, 6ispridlc 36558 . . . . . 6 (𝑅 ∈ CRingOps β†’ ({𝑍} ∈ (PrIdlβ€˜π‘…) ↔ ({𝑍} ∈ (Idlβ€˜π‘…) ∧ {𝑍} β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
8 crngorngo 36488 . . . . . . 7 (𝑅 ∈ CRingOps β†’ 𝑅 ∈ RingOps)
98biantrurd 534 . . . . . 6 (𝑅 ∈ CRingOps β†’ ({𝑍} ∈ (PrIdlβ€˜π‘…) ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdlβ€˜π‘…))))
10 3anass 1096 . . . . . . 7 (({𝑍} ∈ (Idlβ€˜π‘…) ∧ {𝑍} β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ ({𝑍} ∈ (Idlβ€˜π‘…) ∧ ({𝑍} β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))))
112, 30idl 36513 . . . . . . . . . 10 (𝑅 ∈ RingOps β†’ {𝑍} ∈ (Idlβ€˜π‘…))
128, 11syl 17 . . . . . . . . 9 (𝑅 ∈ CRingOps β†’ {𝑍} ∈ (Idlβ€˜π‘…))
1312biantrurd 534 . . . . . . . 8 (𝑅 ∈ CRingOps β†’ (({𝑍} β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ ({𝑍} ∈ (Idlβ€˜π‘…) ∧ ({𝑍} β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))))))
142rneqi 5897 . . . . . . . . . . . . . . 15 ran 𝐺 = ran (1st β€˜π‘…)
156, 14eqtri 2765 . . . . . . . . . . . . . 14 𝑋 = ran (1st β€˜π‘…)
16 isdmn3.5 . . . . . . . . . . . . . 14 π‘ˆ = (GIdβ€˜π»)
1715, 5, 16rngo1cl 36427 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps β†’ π‘ˆ ∈ 𝑋)
18 eleq2 2827 . . . . . . . . . . . . . 14 ({𝑍} = 𝑋 β†’ (π‘ˆ ∈ {𝑍} ↔ π‘ˆ ∈ 𝑋))
19 elsni 4608 . . . . . . . . . . . . . 14 (π‘ˆ ∈ {𝑍} β†’ π‘ˆ = 𝑍)
2018, 19syl6bir 254 . . . . . . . . . . . . 13 ({𝑍} = 𝑋 β†’ (π‘ˆ ∈ 𝑋 β†’ π‘ˆ = 𝑍))
2117, 20syl5com 31 . . . . . . . . . . . 12 (𝑅 ∈ RingOps β†’ ({𝑍} = 𝑋 β†’ π‘ˆ = 𝑍))
222, 5, 3, 16, 6rngoueqz 36428 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps β†’ (𝑋 β‰ˆ 1o ↔ π‘ˆ = 𝑍))
232, 6, 3rngo0cl 36407 . . . . . . . . . . . . . 14 (𝑅 ∈ RingOps β†’ 𝑍 ∈ 𝑋)
24 en1eqsn 9225 . . . . . . . . . . . . . . . 16 ((𝑍 ∈ 𝑋 ∧ 𝑋 β‰ˆ 1o) β†’ 𝑋 = {𝑍})
2524eqcomd 2743 . . . . . . . . . . . . . . 15 ((𝑍 ∈ 𝑋 ∧ 𝑋 β‰ˆ 1o) β†’ {𝑍} = 𝑋)
2625ex 414 . . . . . . . . . . . . . 14 (𝑍 ∈ 𝑋 β†’ (𝑋 β‰ˆ 1o β†’ {𝑍} = 𝑋))
2723, 26syl 17 . . . . . . . . . . . . 13 (𝑅 ∈ RingOps β†’ (𝑋 β‰ˆ 1o β†’ {𝑍} = 𝑋))
2822, 27sylbird 260 . . . . . . . . . . . 12 (𝑅 ∈ RingOps β†’ (π‘ˆ = 𝑍 β†’ {𝑍} = 𝑋))
2921, 28impbid 211 . . . . . . . . . . 11 (𝑅 ∈ RingOps β†’ ({𝑍} = 𝑋 ↔ π‘ˆ = 𝑍))
308, 29syl 17 . . . . . . . . . 10 (𝑅 ∈ CRingOps β†’ ({𝑍} = 𝑋 ↔ π‘ˆ = 𝑍))
3130necon3bid 2989 . . . . . . . . 9 (𝑅 ∈ CRingOps β†’ ({𝑍} β‰  𝑋 ↔ π‘ˆ β‰  𝑍))
32 ovex 7395 . . . . . . . . . . . . 13 (π‘Žπ»π‘) ∈ V
3332elsn 4606 . . . . . . . . . . . 12 ((π‘Žπ»π‘) ∈ {𝑍} ↔ (π‘Žπ»π‘) = 𝑍)
34 velsn 4607 . . . . . . . . . . . . 13 (π‘Ž ∈ {𝑍} ↔ π‘Ž = 𝑍)
35 velsn 4607 . . . . . . . . . . . . 13 (𝑏 ∈ {𝑍} ↔ 𝑏 = 𝑍)
3634, 35orbi12i 914 . . . . . . . . . . . 12 ((π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}) ↔ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍))
3733, 36imbi12i 351 . . . . . . . . . . 11 (((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍)))
3837a1i 11 . . . . . . . . . 10 (𝑅 ∈ CRingOps β†’ (((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍))))
39382ralbidv 3213 . . . . . . . . 9 (𝑅 ∈ CRingOps β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})) ↔ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍))))
4031, 39anbi12d 632 . . . . . . . 8 (𝑅 ∈ CRingOps β†’ (({𝑍} β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ (π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍)))))
4113, 40bitr3d 281 . . . . . . 7 (𝑅 ∈ CRingOps β†’ (({𝑍} ∈ (Idlβ€˜π‘…) ∧ ({𝑍} β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍})))) ↔ (π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍)))))
4210, 41bitrid 283 . . . . . 6 (𝑅 ∈ CRingOps β†’ (({𝑍} ∈ (Idlβ€˜π‘…) ∧ {𝑍} β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ {𝑍} β†’ (π‘Ž ∈ {𝑍} ∨ 𝑏 ∈ {𝑍}))) ↔ (π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍)))))
437, 9, 423bitr3d 309 . . . . 5 (𝑅 ∈ CRingOps β†’ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdlβ€˜π‘…)) ↔ (π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍)))))
444, 43bitrid 283 . . . 4 (𝑅 ∈ CRingOps β†’ (𝑅 ∈ PrRing ↔ (π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍)))))
4544pm5.32i 576 . . 3 ((𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing) ↔ (𝑅 ∈ CRingOps ∧ (π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍)))))
46 ancom 462 . . 3 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ 𝑅 ∈ PrRing))
47 3anass 1096 . . 3 ((𝑅 ∈ CRingOps ∧ π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍))) ↔ (𝑅 ∈ CRingOps ∧ (π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍)))))
4845, 46, 473bitr4i 303 . 2 ((𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps) ↔ (𝑅 ∈ CRingOps ∧ π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍))))
491, 48bitri 275 1 (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  {csn 4591   class class class wbr 5110  ran crn 5639  β€˜cfv 6501  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  1oc1o 8410   β‰ˆ cen 8887  GIdcgi 29474  RingOpscrngo 36382  CRingOpsccring 36481  Idlcidl 36495  PrIdlcpridl 36496  PrRingcprrng 36534  Dmncdmn 36535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-1o 8417  df-en 8891  df-grpo 29477  df-gid 29478  df-ginv 29479  df-ablo 29529  df-ass 36331  df-exid 36333  df-mgmOLD 36337  df-sgrOLD 36349  df-mndo 36355  df-rngo 36383  df-com2 36478  df-crngo 36482  df-idl 36498  df-pridl 36499  df-prrngo 36536  df-dmn 36537  df-igen 36548
This theorem is referenced by:  dmnnzd  36563
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