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Theorem isdmn3 30633
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 30614 . 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 30609 . . . . 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 30629 . . . . . 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 30559 . . . . . . 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 30584 . . . . . . . . . 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 5239 . . . . . . . . . . . . . . 15  |-  ran  G  =  ran  ( 1st `  R
)
156, 14eqtri 2486 . . . . . . . . . . . . . 14  |-  X  =  ran  ( 1st `  R
)
16 isdmn3.5 . . . . . . . . . . . . . 14  |-  U  =  (GId `  H )
1715, 5, 16rngo1cl 25557 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  U  e.  X
)
18 eleq2 2530 . . . . . . . . . . . . . 14  |-  ( { Z }  =  X  ->  ( U  e. 
{ Z }  <->  U  e.  X ) )
19 elsni 4057 . . . . . . . . . . . . . 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 25558 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  U  =  Z ) )
232, 6, 3rngo0cl 25526 . . . . . . . . . . . . . 14  |-  ( R  e.  RingOps  ->  Z  e.  X
)
24 en1eqsn 7768 . . . . . . . . . . . . . . . 16  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  X  =  { Z } )
2524eqcomd 2465 . . . . . . . . . . . . . . 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 2715 . . . . . . . . 9  |-  ( R  e. CRingOps  ->  ( { Z }  =/=  X  <->  U  =/=  Z ) )
32 ovex 6324 . . . . . . . . . . . . 13  |-  ( a H b )  e. 
_V
3332elsnc 4056 . . . . . . . . . . . 12  |-  ( ( a H b )  e.  { Z }  <->  ( a H b )  =  Z )
34 elsn 4046 . . . . . . . . . . . . 13  |-  ( a  e.  { Z }  <->  a  =  Z )
35 elsn 4046 . . . . . . . . . . . . 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 2901 . . . . . . . . 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   {csn 4032   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   1oc1o 7141    ~~ cen 7532  GIdcgi 25315   RingOpscrngo 25503  CRingOpsccring 30554   Idlcidl 30566   PrIdlcpridl 30567   PrRingcprrng 30605   Dmncdmn 30606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-grpo 25319  df-gid 25320  df-ginv 25321  df-ablo 25410  df-ass 25441  df-exid 25443  df-mgmOLD 25447  df-sgrOLD 25459  df-mndo 25466  df-rngo 25504  df-com2 25539  df-crngo 30555  df-idl 30569  df-pridl 30570  df-prrngo 30607  df-dmn 30608  df-igen 30619
This theorem is referenced by:  dmnnzd  30634
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