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Theorem isdmn3 26574
Description: The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
isdmn3.1  |-  G  =  ( 1st `  R
)
isdmn3.2  |-  H  =  ( 2nd `  R
)
isdmn3.3  |-  X  =  ran  G
isdmn3.4  |-  Z  =  (GId `  G )
isdmn3.5  |-  U  =  (GId `  H )
Assertion
Ref Expression
isdmn3  |-  ( R  e.  Dmn  <->  ( R  e. CRingOps 
/\  U  =/=  Z  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  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 26555 . 2  |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )
2 isdmn3.1 . . . . . 6  |-  G  =  ( 1st `  R
)
3 isdmn3.4 . . . . . 6  |-  Z  =  (GId `  G )
42, 3isprrngo 26550 . . . . 5  |-  ( R  e.  PrRing 
<->  ( R  e.  RingOps  /\  { Z }  e.  (
PrIdl `  R ) ) )
5 isdmn3.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
6 isdmn3.3 . . . . . . 7  |-  X  =  ran  G
72, 5, 6ispridlc 26570 . . . . . 6  |-  ( R  e. CRingOps  ->  ( { Z }  e.  ( PrIdl `  R )  <->  ( { Z }  e.  ( Idl `  R )  /\  { Z }  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  { Z }  ->  ( a  e.  { Z }  \/  b  e.  { Z } ) ) ) ) )
8 crngorngo 26500 . . . . . . 7  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
98biantrurd 495 . . . . . 6  |-  ( R  e. CRingOps  ->  ( { Z }  e.  ( PrIdl `  R )  <->  ( R  e.  RingOps  /\  { Z }  e.  ( PrIdl `  R ) ) ) )
10 3anass 940 . . . . . . 7  |-  ( ( { Z }  e.  ( Idl `  R )  /\  { Z }  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( (
a H b )  e.  { Z }  ->  ( a  e.  { Z }  \/  b  e.  { Z } ) ) )  <->  ( { Z }  e.  ( Idl `  R )  /\  ( { Z }  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  e. 
{ Z }  ->  ( a  e.  { Z }  \/  b  e.  { Z } ) ) ) ) )
112, 30idl 26525 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
128, 11syl 16 . . . . . . . . 9  |-  ( R  e. CRingOps  ->  { Z }  e.  ( Idl `  R
) )
1312biantrurd 495 . . . . . . . 8  |-  ( R  e. CRingOps  ->  ( ( { Z }  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  { Z }  ->  ( a  e.  { Z }  \/  b  e.  { Z } ) ) )  <-> 
( { Z }  e.  ( Idl `  R
)  /\  ( { Z }  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  { Z }  ->  ( a  e.  { Z }  \/  b  e.  { Z } ) ) ) ) ) )
142rneqi 5055 . . . . . . . . . . . . . . 15  |-  ran  G  =  ran  ( 1st `  R
)
156, 14eqtri 2424 . . . . . . . . . . . . . 14  |-  X  =  ran  ( 1st `  R
)
16 isdmn3.5 . . . . . . . . . . . . . 14  |-  U  =  (GId `  H )
1715, 5, 16rngo1cl 21970 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  U  e.  X
)
18 eleq2 2465 . . . . . . . . . . . . . 14  |-  ( { Z }  =  X  ->  ( U  e. 
{ Z }  <->  U  e.  X ) )
19 elsni 3798 . . . . . . . . . . . . . 14  |-  ( U  e.  { Z }  ->  U  =  Z )
2018, 19syl6bir 221 . . . . . . . . . . . . 13  |-  ( { Z }  =  X  ->  ( U  e.  X  ->  U  =  Z ) )
2117, 20syl5com 28 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  ( { Z }  =  X  ->  U  =  Z ) )
222, 5, 3, 16, 6rngoueqz 21971 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  U  =  Z ) )
232, 6, 3rngo0cl 21939 . . . . . . . . . . . . . 14  |-  ( R  e.  RingOps  ->  Z  e.  X
)
24 en1eqsn 7297 . . . . . . . . . . . . . . . 16  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  X  =  { Z } )
2524eqcomd 2409 . . . . . . . . . . . . . . 15  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  { Z }  =  X )
2625ex 424 . . . . . . . . . . . . . 14  |-  ( Z  e.  X  ->  ( X  ~~  1o  ->  { Z }  =  X )
)
2723, 26syl 16 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  { Z }  =  X ) )
2822, 27sylbird 227 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  ( U  =  Z  ->  { Z }  =  X )
)
2921, 28impbid 184 . . . . . . . . . . 11  |-  ( R  e.  RingOps  ->  ( { Z }  =  X  <->  U  =  Z ) )
308, 29syl 16 . . . . . . . . . 10  |-  ( R  e. CRingOps  ->  ( { Z }  =  X  <->  U  =  Z ) )
3130necon3bid 2602 . . . . . . . . 9  |-  ( R  e. CRingOps  ->  ( { Z }  =/=  X  <->  U  =/=  Z ) )
32 ovex 6065 . . . . . . . . . . . . 13  |-  ( a H b )  e. 
_V
3332elsnc 3797 . . . . . . . . . . . 12  |-  ( ( a H b )  e.  { Z }  <->  ( a H b )  =  Z )
34 elsn 3789 . . . . . . . . . . . . 13  |-  ( a  e.  { Z }  <->  a  =  Z )
35 elsn 3789 . . . . . . . . . . . . 13  |-  ( b  e.  { Z }  <->  b  =  Z )
3634, 35orbi12i 508 . . . . . . . . . . . 12  |-  ( ( a  e.  { Z }  \/  b  e.  { Z } )  <->  ( a  =  Z  \/  b  =  Z ) )
3733, 36imbi12i 317 . . . . . . . . . . 11  |-  ( ( ( a H b )  e.  { Z }  ->  ( a  e. 
{ Z }  \/  b  e.  { Z } ) )  <->  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) )
3837a1i 11 . . . . . . . . . 10  |-  ( R  e. CRingOps  ->  ( ( ( a H b )  e.  { Z }  ->  ( a  e.  { Z }  \/  b  e.  { Z } ) )  <->  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) ) ) )
39382ralbidv 2708 . . . . . . . . 9  |-  ( R  e. CRingOps  ->  ( A. a  e.  X  A. b  e.  X  ( (
a H b )  e.  { Z }  ->  ( a  e.  { Z }  \/  b  e.  { Z } ) )  <->  A. a  e.  X  A. b  e.  X  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) ) ) )
4031, 39anbi12d 692 . . . . . . . 8  |-  ( R  e. CRingOps  ->  ( ( { Z }  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  e.  { Z }  ->  ( a  e.  { Z }  \/  b  e.  { Z } ) ) )  <-> 
( U  =/=  Z  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) ) ) ) )
4113, 40bitr3d 247 . . . . . . 7  |-  ( R  e. CRingOps  ->  ( ( { Z }  e.  ( Idl `  R )  /\  ( { Z }  =/=  X  /\  A. a  e.  X  A. b  e.  X  (
( a H b )  e.  { Z }  ->  ( a  e. 
{ Z }  \/  b  e.  { Z } ) ) ) )  <->  ( U  =/= 
Z  /\  A. a  e.  X  A. b  e.  X  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) ) ) )
4210, 41syl5bb 249 . . . . . 6  |-  ( R  e. CRingOps  ->  ( ( { Z }  e.  ( Idl `  R )  /\  { Z }  =/=  X  /\  A. a  e.  X  A. b  e.  X  ( (
a H b )  e.  { Z }  ->  ( a  e.  { Z }  \/  b  e.  { Z } ) ) )  <->  ( U  =/=  Z  /\  A. a  e.  X  A. b  e.  X  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) ) ) )
437, 9, 423bitr3d 275 . . . . 5  |-  ( R  e. CRingOps  ->  ( ( R  e.  RingOps  /\  { Z }  e.  ( PrIdl `  R ) )  <->  ( U  =/=  Z  /\  A. a  e.  X  A. b  e.  X  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) ) ) )
444, 43syl5bb 249 . . . 4  |-  ( R  e. CRingOps  ->  ( R  e. 
PrRing 
<->  ( U  =/=  Z  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) ) ) ) )
4544pm5.32i 619 . . 3  |-  ( ( R  e. CRingOps  /\  R  e. 
PrRing )  <->  ( R  e. CRingOps  /\  ( U  =/=  Z  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) ) ) ) )
46 ancom 438 . . 3  |-  ( ( R  e.  PrRing  /\  R  e. CRingOps )  <->  ( R  e. CRingOps  /\  R  e.  PrRing ) )
47 3anass 940 . . 3  |-  ( ( R  e. CRingOps  /\  U  =/= 
Z  /\  A. a  e.  X  A. b  e.  X  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) )  <->  ( R  e. CRingOps 
/\  ( U  =/= 
Z  /\  A. a  e.  X  A. b  e.  X  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) ) ) )
4845, 46, 473bitr4i 269 . 2  |-  ( ( R  e.  PrRing  /\  R  e. CRingOps )  <->  ( R  e. CRingOps  /\  U  =/=  Z  /\  A. a  e.  X  A. b  e.  X  (
( a H b )  =  Z  -> 
( a  =  Z  \/  b  =  Z ) ) ) )
491, 48bitri 241 1  |-  ( R  e.  Dmn  <->  ( R  e. CRingOps 
/\  U  =/=  Z  /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   {csn 3774   class class class wbr 4172   ran crn 4838   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   1oc1o 6676    ~~ cen 7065  GIdcgi 21728   RingOpscrngo 21916  CRingOpsccring 26495   Idlcidl 26507   PrIdlcpridl 26508   PrRingcprrng 26546   Dmncdmn 26547
This theorem is referenced by:  dmnnzd  26575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-grpo 21732  df-gid 21733  df-ginv 21734  df-ablo 21823  df-ass 21854  df-exid 21856  df-mgm 21860  df-sgr 21872  df-mndo 21879  df-rngo 21917  df-com2 21952  df-crngo 26496  df-idl 26510  df-pridl 26511  df-prrngo 26548  df-dmn 26549  df-igen 26560
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