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Theorem isdmn3 26205
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 26186 . 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 26181 . . . . 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 26201 . . . . . 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 26131 . . . . . . 7  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
98biantrurd 494 . . . . . 6  |-  ( R  e. CRingOps  ->  ( { Z }  e.  ( PrIdl `  R )  <->  ( R  e.  RingOps  /\  { Z }  e.  ( PrIdl `  R ) ) ) )
10 3anass 939 . . . . . . 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 26156 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
128, 11syl 15 . . . . . . . . 9  |-  ( R  e. CRingOps  ->  { Z }  e.  ( Idl `  R
) )
1312biantrurd 494 . . . . . . . 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 5008 . . . . . . . . . . . . . . 15  |-  ran  G  =  ran  ( 1st `  R
)
156, 14eqtri 2386 . . . . . . . . . . . . . 14  |-  X  =  ran  ( 1st `  R
)
16 isdmn3.5 . . . . . . . . . . . . . 14  |-  U  =  (GId `  H )
1715, 5, 16rngo1cl 21407 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  U  e.  X
)
18 eleq2 2427 . . . . . . . . . . . . . 14  |-  ( { Z }  =  X  ->  ( U  e. 
{ Z }  <->  U  e.  X ) )
19 elsni 3753 . . . . . . . . . . . . . 14  |-  ( U  e.  { Z }  ->  U  =  Z )
2018, 19syl6bir 220 . . . . . . . . . . . . 13  |-  ( { Z }  =  X  ->  ( U  e.  X  ->  U  =  Z ) )
2117, 20syl5com 26 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  ( { Z }  =  X  ->  U  =  Z ) )
222, 5, 3, 16, 6rngoueqz 21408 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( X  ~~  1o 
<->  U  =  Z ) )
232, 6, 3rngo0cl 21376 . . . . . . . . . . . . . 14  |-  ( R  e.  RingOps  ->  Z  e.  X
)
24 en1eqsn 7235 . . . . . . . . . . . . . . . 16  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  X  =  { Z } )
2524eqcomd 2371 . . . . . . . . . . . . . . 15  |-  ( ( Z  e.  X  /\  X  ~~  1o )  ->  { Z }  =  X )
2625ex 423 . . . . . . . . . . . . . 14  |-  ( Z  e.  X  ->  ( X  ~~  1o  ->  { Z }  =  X )
)
2723, 26syl 15 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( X  ~~  1o  ->  { Z }  =  X ) )
2822, 27sylbird 226 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  ( U  =  Z  ->  { Z }  =  X )
)
2921, 28impbid 183 . . . . . . . . . . 11  |-  ( R  e.  RingOps  ->  ( { Z }  =  X  <->  U  =  Z ) )
308, 29syl 15 . . . . . . . . . 10  |-  ( R  e. CRingOps  ->  ( { Z }  =  X  <->  U  =  Z ) )
3130necon3bid 2564 . . . . . . . . 9  |-  ( R  e. CRingOps  ->  ( { Z }  =/=  X  <->  U  =/=  Z ) )
32 ovex 6006 . . . . . . . . . . . . 13  |-  ( a H b )  e. 
_V
3332elsnc 3752 . . . . . . . . . . . 12  |-  ( ( a H b )  e.  { Z }  <->  ( a H b )  =  Z )
34 elsn 3744 . . . . . . . . . . . . 13  |-  ( a  e.  { Z }  <->  a  =  Z )
35 elsn 3744 . . . . . . . . . . . . 13  |-  ( b  e.  { Z }  <->  b  =  Z )
3634, 35orbi12i 507 . . . . . . . . . . . 12  |-  ( ( a  e.  { Z }  \/  b  e.  { Z } )  <->  ( a  =  Z  \/  b  =  Z ) )
3733, 36imbi12i 316 . . . . . . . . . . 11  |-  ( ( ( a H b )  e.  { Z }  ->  ( a  e. 
{ Z }  \/  b  e.  { Z } ) )  <->  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) )
3837a1i 10 . . . . . . . . . 10  |-  ( R  e. CRingOps  ->  ( ( ( a H b )  e.  { Z }  ->  ( a  e.  { Z }  \/  b  e.  { Z } ) )  <->  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) ) ) )
39382ralbidv 2670 . . . . . . . . 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 691 . . . . . . . 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 246 . . . . . . 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 248 . . . . . 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 274 . . . . 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 248 . . . 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 618 . . 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 437 . . 3  |-  ( ( R  e.  PrRing  /\  R  e. CRingOps )  <->  ( R  e. CRingOps  /\  R  e.  PrRing ) )
47 3anass 939 . . 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 268 . 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 240 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 176    \/ wo 357    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   {csn 3729   class class class wbr 4125   ran crn 4793   ` cfv 5358  (class class class)co 5981   1stc1st 6247   2ndc2nd 6248   1oc1o 6614    ~~ cen 7003  GIdcgi 21165   RingOpscrngo 21353  CRingOpsccring 26126   Idlcidl 26138   PrIdlcpridl 26139   PrRingcprrng 26177   Dmncdmn 26178
This theorem is referenced by:  dmnnzd  26206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-1o 6621  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-grpo 21169  df-gid 21170  df-ginv 21171  df-ablo 21260  df-ass 21291  df-exid 21293  df-mgm 21297  df-sgr 21309  df-mndo 21316  df-rngo 21354  df-com2 21389  df-crngo 26127  df-idl 26141  df-pridl 26142  df-prrngo 26179  df-dmn 26180  df-igen 26191
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