Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isdmn3 Structured version   Unicode version

Theorem isdmn3 30102
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 30083 . 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 30078 . . . . 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 30098 . . . . . 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 30028 . . . . . . 7  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
98biantrurd 508 . . . . . 6  |-  ( R  e. CRingOps  ->  ( { Z }  e.  ( PrIdl `  R )  <->  ( R  e.  RingOps  /\  { Z }  e.  ( PrIdl `  R ) ) ) )
10 3anass 977 . . . . . . 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 30053 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
128, 11syl 16 . . . . . . . . 9  |-  ( R  e. CRingOps  ->  { Z }  e.  ( Idl `  R
) )
1312biantrurd 508 . . . . . . . 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 5229 . . . . . . . . . . . . . . 15  |-  ran  G  =  ran  ( 1st `  R
)
156, 14eqtri 2496 . . . . . . . . . . . . . 14  |-  X  =  ran  ( 1st `  R
)
16 isdmn3.5 . . . . . . . . . . . . . 14  |-  U  =  (GId `  H )
1715, 5, 16rngo1cl 25135 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  U  e.  X
)
18 eleq2 2540 . . . . . . . . . . . . . 14  |-  ( { Z }  =  X  ->  ( U  e. 
{ Z }  <->  U  e.  X ) )
19 elsni 4052 . . . . . . . . . . . . . 14  |-  ( U  e.  { Z }  ->  U  =  Z )
2018, 19syl6bir 229 . . . . . . . . . . . . 13  |-  ( { Z }  =  X  ->  ( U  e.  X  ->  U  =  Z ) )
2117, 20syl5com 30 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  ( { Z }  =  X  ->  U  =  Z ) )
222, 5, 3, 16, 6rngoueqz 25136 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  U  =  Z ) )
232, 6, 3rngo0cl 25104 . . . . . . . . . . . . . 14  |-  ( R  e.  RingOps  ->  Z  e.  X
)
24 en1eqsn 7749 . . . . . . . . . . . . . . . 16  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  X  =  { Z } )
2524eqcomd 2475 . . . . . . . . . . . . . . 15  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  { Z }  =  X )
2625ex 434 . . . . . . . . . . . . . 14  |-  ( Z  e.  X  ->  ( X  ~~  1o  ->  { Z }  =  X )
)
2723, 26syl 16 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  { Z }  =  X ) )
2822, 27sylbird 235 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  ( U  =  Z  ->  { Z }  =  X )
)
2921, 28impbid 191 . . . . . . . . . . 11  |-  ( R  e.  RingOps  ->  ( { Z }  =  X  <->  U  =  Z ) )
308, 29syl 16 . . . . . . . . . 10  |-  ( R  e. CRingOps  ->  ( { Z }  =  X  <->  U  =  Z ) )
3130necon3bid 2725 . . . . . . . . 9  |-  ( R  e. CRingOps  ->  ( { Z }  =/=  X  <->  U  =/=  Z ) )
32 ovex 6309 . . . . . . . . . . . . 13  |-  ( a H b )  e. 
_V
3332elsnc 4051 . . . . . . . . . . . 12  |-  ( ( a H b )  e.  { Z }  <->  ( a H b )  =  Z )
34 elsn 4041 . . . . . . . . . . . . 13  |-  ( a  e.  { Z }  <->  a  =  Z )
35 elsn 4041 . . . . . . . . . . . . 13  |-  ( b  e.  { Z }  <->  b  =  Z )
3634, 35orbi12i 521 . . . . . . . . . . . 12  |-  ( ( a  e.  { Z }  \/  b  e.  { Z } )  <->  ( a  =  Z  \/  b  =  Z ) )
3733, 36imbi12i 326 . . . . . . . . . . 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 2908 . . . . . . . . 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 710 . . . . . . . 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 255 . . . . . . 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 257 . . . . . 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 283 . . . . 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 257 . . . 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 637 . . 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 450 . . 3  |-  ( ( R  e.  PrRing  /\  R  e. CRingOps )  <->  ( R  e. CRingOps  /\  R  e.  PrRing ) )
47 3anass 977 . . 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 277 . 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 249 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {csn 4027   class class class wbr 4447   ran crn 5000   ` cfv 5588  (class class class)co 6284   1stc1st 6782   2ndc2nd 6783   1oc1o 7123    ~~ cen 7513  GIdcgi 24893   RingOpscrngo 25081  CRingOpsccring 30023   Idlcidl 30035   PrIdlcpridl 30036   PrRingcprrng 30074   Dmncdmn 30075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-grpo 24897  df-gid 24898  df-ginv 24899  df-ablo 24988  df-ass 25019  df-exid 25021  df-mgm 25025  df-sgr 25037  df-mndo 25044  df-rngo 25082  df-com2 25117  df-crngo 30024  df-idl 30038  df-pridl 30039  df-prrngo 30076  df-dmn 30077  df-igen 30088
This theorem is referenced by:  dmnnzd  30103
  Copyright terms: Public domain W3C validator