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

Theorem isdmn3 26578
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 26559 . 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 26554 . . . . 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 26574 . . . . . 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 26504 . . . . . . 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 26529 . . . . . . . . . 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 5059 . . . . . . . . . . . . . . 15  |-  ran  G  =  ran  ( 1st `  R
)
156, 14eqtri 2428 . . . . . . . . . . . . . 14  |-  X  =  ran  ( 1st `  R
)
16 isdmn3.5 . . . . . . . . . . . . . 14  |-  U  =  (GId `  H )
1715, 5, 16rngo1cl 21974 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  U  e.  X
)
18 eleq2 2469 . . . . . . . . . . . . . 14  |-  ( { Z }  =  X  ->  ( U  e. 
{ Z }  <->  U  e.  X ) )
19 elsni 3802 . . . . . . . . . . . . . 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 21975 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  U  =  Z ) )
232, 6, 3rngo0cl 21943 . . . . . . . . . . . . . 14  |-  ( R  e.  RingOps  ->  Z  e.  X
)
24 en1eqsn 7301 . . . . . . . . . . . . . . . 16  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  X  =  { Z } )
2524eqcomd 2413 . . . . . . . . . . . . . . 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 2606 . . . . . . . . 9  |-  ( R  e. CRingOps  ->  ( { Z }  =/=  X  <->  U  =/=  Z ) )
32 ovex 6069 . . . . . . . . . . . . 13  |-  ( a H b )  e. 
_V
3332elsnc 3801 . . . . . . . . . . . 12  |-  ( ( a H b )  e.  { Z }  <->  ( a H b )  =  Z )
34 elsn 3793 . . . . . . . . . . . . 13  |-  ( a  e.  { Z }  <->  a  =  Z )
35 elsn 3793 . . . . . . . . . . . . 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 2712 . . . . . . . . 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 2571   A.wral 2670   {csn 3778   class class class wbr 4176   ran crn 4842   ` cfv 5417  (class class class)co 6044   1stc1st 6310   2ndc2nd 6311   1oc1o 6680    ~~ cen 7069  GIdcgi 21732   RingOpscrngo 21920  CRingOpsccring 26499   Idlcidl 26511   PrIdlcpridl 26512   PrRingcprrng 26550   Dmncdmn 26551
This theorem is referenced by:  dmnnzd  26579
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-1o 6687  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-grpo 21736  df-gid 21737  df-ginv 21738  df-ablo 21827  df-ass 21858  df-exid 21860  df-mgm 21864  df-sgr 21876  df-mndo 21883  df-rngo 21921  df-com2 21956  df-crngo 26500  df-idl 26514  df-pridl 26515  df-prrngo 26552  df-dmn 26553  df-igen 26564
  Copyright terms: Public domain W3C validator