Theorem isdrngo1 33755

Description: The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)

Hypotheses

Ref	Expression
isdivrng1.1	|- G = ( 1st ` R )
isdivrng1.2	|- H = ( 2nd ` R )
isdivrng1.3	|- Z = (GId ` G )
isdivrng1.4	|- X = ran G

Assertion

Ref	Expression
isdrngo1	|- ( R e. DivRingOps <-> ( R e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) )

Proof of Theorem isdrngo1

Dummy variables g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-drngo 33748	. . . 4 |- DivRingOps = { <. g , h >. | ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) }
2	1	relopabi 5245	. . 3 |- Rel DivRingOps
3		1st2nd 7214	. . 3 |- ( ( Rel DivRingOps /\ R e. DivRingOps ) -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
4	2, 3	mpan 706	. 2 |- ( R e. DivRingOps -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
5		relrngo 33695	. . . 4 |- Rel RingOps
6		1st2nd 7214	. . . 4 |- ( ( Rel RingOps /\ R e. RingOps ) -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
7	5, 6	mpan 706	. . 3 |- ( R e. RingOps -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
8	7	adantr 481	. 2 |- ( ( R e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
9		isdivrng1.1	. . . . 5 |- G = ( 1st ` R )
10		isdivrng1.2	. . . . 5 |- H = ( 2nd ` R )
11	9, 10	opeq12i 4407	. . . 4 |- <. G , H >. = <. ( 1st ` R ) , ( 2nd ` R ) >.
12	11	eqeq2i 2634	. . 3 |- ( R = <. G , H >. <-> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
13		fvex 6201	. . . . . . 7 |- ( 2nd ` R ) e. _V
14	10, 13	eqeltri 2697	. . . . . 6 |- H e. _V
15		isdivrngo 33749	. . . . . 6 |- ( H e. _V -> ( <. G , H >. e. DivRingOps <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) ) )
16	14, 15	ax-mp 5	. . . . 5 |- ( <. G , H >. e. DivRingOps <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) )
17		isdivrng1.4	. . . . . . . . . 10 |- X = ran G
18		isdivrng1.3	. . . . . . . . . . 11 |- Z = (GId ` G )
19	18	sneqi 4188	. . . . . . . . . 10 |- { Z } = { (GId ` G ) }
20	17, 19	difeq12i 3726	. . . . . . . . 9 |- ( X \ { Z } ) = ( ran G \ { (GId ` G ) } )
21	20, 20	xpeq12i 5137	. . . . . . . 8 |- ( ( X \ { Z } ) X. ( X \ { Z } ) ) = ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) )
22	21	reseq2i 5393	. . . . . . 7 |- ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) = ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) )
23	22	eleq1i 2692	. . . . . 6 |- ( ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp <-> ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp )
24	23	anbi2i 730	. . . . 5 |- ( ( <. G , H >. e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) )
25	16, 24	bitr4i 267	. . . 4 |- ( <. G , H >. e. DivRingOps <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) )
26		eleq1 2689	. . . . 5 |- ( R = <. G , H >. -> ( R e. DivRingOps <-> <. G , H >. e. DivRingOps ) )
27		eleq1 2689	. . . . . 6 |- ( R = <. G , H >. -> ( R e. RingOps <-> <. G , H >. e. RingOps ) )
28	27	anbi1d 741	. . . . 5 |- ( R = <. G , H >. -> ( ( R e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) ) )
29	26, 28	bibi12d 335	. . . 4 |- ( R = <. G , H >. -> ( ( R e. DivRingOps <-> ( R e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) ) <-> ( <. G , H >. e. DivRingOps <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) ) ) )
30	25, 29	mpbiri 248	. . 3 |- ( R = <. G , H >. -> ( R e. DivRingOps <-> ( R e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) ) )
31	12, 30	sylbir 225	. 2 |- ( R = <. ( 1st ` R ) , ( 2nd ` R ) >. -> ( R e. DivRingOps <-> ( R e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) ) )
32	4, 8, 31	pm5.21nii 368	1 |- ( R e. DivRingOps <-> ( R e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   {csn 4177   <.cop 4183   X. cxp 5112   ran crn 5115   |` cres 5116   Rel wrel 5119   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   GrpOpcgr 27343  GIdcgi 27344   RingOpscrngo 33693   DivRingOpscdrng 33747

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-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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-rngo 33694  df-drngo 33748

This theorem is referenced by:  divrngcl  33756  isdrngo2  33757  divrngpr  33852