Theorem isdmn3 33873

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

Step	Hyp	Ref	Expression
1		isdmn2 33854	. 2 |- ( R e. Dmn <-> ( R e. PrRing /\ R e. CRingOps ) )
2		isdmn3.1	. . . . . 6 |- G = ( 1st ` R )
3		isdmn3.4	. . . . . 6 |- Z = (GId ` G )
4	2, 3	isprrngo 33849	. . . . 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
7	2, 5, 6	ispridlc 33869	. . . . . 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 33799	. . . . . . 7 |- ( R e. CRingOps -> R e. RingOps )
9	8	biantrurd 529	. . . . . 6 |- ( R e. CRingOps -> ( { Z } e. ( PrIdl ` R ) <-> ( R e. RingOps /\ { Z } e. ( PrIdl ` R ) ) ) )
10		3anass 1042	. . . . . . 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 } ) ) ) ) )
11	2, 3	0idl 33824	. . . . . . . . . 10 |- ( R e. RingOps -> { Z } e. ( Idl ` R ) )
12	8, 11	syl 17	. . . . . . . . 9 |- ( R e. CRingOps -> { Z } e. ( Idl ` R ) )
13	12	biantrurd 529	. . . . . . . 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 } ) ) ) ) ) )
14	2	rneqi 5352	. . . . . . . . . . . . . . 15 |- ran G = ran ( 1st ` R )
15	6, 14	eqtri 2644	. . . . . . . . . . . . . 14 |- X = ran ( 1st ` R )
16		isdmn3.5	. . . . . . . . . . . . . 14 |- U = (GId ` H )
17	15, 5, 16	rngo1cl 33738	. . . . . . . . . . . . 13 |- ( R e. RingOps -> U e. X )
18		eleq2 2690	. . . . . . . . . . . . . 14 |- ( { Z } = X -> ( U e. { Z } <-> U e. X ) )
19		elsni 4194	. . . . . . . . . . . . . 14 |- ( U e. { Z } -> U = Z )
20	18, 19	syl6bir 244	. . . . . . . . . . . . 13 |- ( { Z } = X -> ( U e. X -> U = Z ) )
21	17, 20	syl5com 31	. . . . . . . . . . . 12 |- ( R e. RingOps -> ( { Z } = X -> U = Z ) )
22	2, 5, 3, 16, 6	rngoueqz 33739	. . . . . . . . . . . . 13 |- ( R e. RingOps -> ( X ~~ 1o <-> U = Z ) )
23	2, 6, 3	rngo0cl 33718	. . . . . . . . . . . . . 14 |- ( R e. RingOps -> Z e. X )
24		en1eqsn 8190	. . . . . . . . . . . . . . . 16 |- ( ( Z e. X /\ X ~~ 1o ) -> X = { Z } )
25	24	eqcomd 2628	. . . . . . . . . . . . . . 15 |- ( ( Z e. X /\ X ~~ 1o ) -> { Z } = X )
26	25	ex 450	. . . . . . . . . . . . . 14 |- ( Z e. X -> ( X ~~ 1o -> { Z } = X ) )
27	23, 26	syl 17	. . . . . . . . . . . . 13 |- ( R e. RingOps -> ( X ~~ 1o -> { Z } = X ) )
28	22, 27	sylbird 250	. . . . . . . . . . . 12 |- ( R e. RingOps -> ( U = Z -> { Z } = X ) )
29	21, 28	impbid 202	. . . . . . . . . . 11 |- ( R e. RingOps -> ( { Z } = X <-> U = Z ) )
30	8, 29	syl 17	. . . . . . . . . 10 |- ( R e. CRingOps -> ( { Z } = X <-> U = Z ) )
31	30	necon3bid 2838	. . . . . . . . 9 |- ( R e. CRingOps -> ( { Z } =/= X <-> U =/= Z ) )
32		ovex 6678	. . . . . . . . . . . . 13 |- ( a H b ) e. _V
33	32	elsn 4192	. . . . . . . . . . . 12 |- ( ( a H b ) e. { Z } <-> ( a H b ) = Z )
34		velsn 4193	. . . . . . . . . . . . 13 |- ( a e. { Z } <-> a = Z )
35		velsn 4193	. . . . . . . . . . . . 13 |- ( b e. { Z } <-> b = Z )
36	34, 35	orbi12i 543	. . . . . . . . . . . 12 |- ( ( a e. { Z } \/ b e. { Z } ) <-> ( a = Z \/ b = Z ) )
37	33, 36	imbi12i 340	. . . . . . . . . . 11 |- ( ( ( a H b ) e. { Z } -> ( a e. { Z } \/ b e. { Z } ) ) <-> ( ( a H b ) = Z -> ( a = Z \/ b = Z ) ) )
38	37	a1i 11	. . . . . . . . . 10 |- ( R e. CRingOps -> ( ( ( a H b ) e. { Z } -> ( a e. { Z } \/ b e. { Z } ) ) <-> ( ( a H b ) = Z -> ( a = Z \/ b = Z ) ) ) )
39	38	2ralbidv 2989	. . . . . . . . 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 ) ) ) )
40	31, 39	anbi12d 747	. . . . . . . 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 ) ) ) ) )
41	13, 40	bitr3d 270	. . . . . . 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 ) ) ) ) )
42	10, 41	syl5bb 272	. . . . . 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 ) ) ) ) )
43	7, 9, 42	3bitr3d 298	. . . . 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 ) ) ) ) )
44	4, 43	syl5bb 272	. . . 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 ) ) ) ) )
45	44	pm5.32i 669	. . 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 466	. . 3 |- ( ( R e. PrRing /\ R e. CRingOps ) <-> ( R e. CRingOps /\ R e. PrRing ) )
47		3anass 1042	. . 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 ) ) ) ) )
48	45, 46, 47	3bitr4i 292	. 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 ) ) ) )
49	1, 48	bitri 264	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 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {csn 4177   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   1oc1o 7553   ~~ cen 7952  GIdcgi 27344   RingOpscrngo 33693  CRingOpsccring 33792   Idlcidl 33806   PrIdlcpridl 33807   PrRingcprrng 33845   Dmncdmn 33846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-com2 33789  df-crngo 33793  df-idl 33809  df-pridl 33810  df-prrngo 33847  df-dmn 33848  df-igen 33859

This theorem is referenced by:  dmnnzd  33874