Theorem drngoi 33750

Description: The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem drngoi

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

Step	Hyp	Ref	Expression
1		opeq1 4402	. . . . . 6 |- ( g = ( 1st ` R ) -> <. g , h >. = <. ( 1st ` R ) , h >. )
2	1	eleq1d 2686	. . . . 5 |- ( g = ( 1st ` R ) -> ( <. g , h >. e. RingOps <-> <. ( 1st ` R ) , h >. e. RingOps ) )
3		id 22	. . . . . . . . . . . 12 |- ( g = ( 1st ` R ) -> g = ( 1st ` R ) )
4		drngi.1	. . . . . . . . . . . 12 |- G = ( 1st ` R )
5	3, 4	syl6eqr 2674	. . . . . . . . . . 11 |- ( g = ( 1st ` R ) -> g = G )
6	5	rneqd 5353	. . . . . . . . . 10 |- ( g = ( 1st ` R ) -> ran g = ran G )
7		drngi.3	. . . . . . . . . 10 |- X = ran G
8	6, 7	syl6eqr 2674	. . . . . . . . 9 |- ( g = ( 1st ` R ) -> ran g = X )
9	5	fveq2d 6195	. . . . . . . . . . 11 |- ( g = ( 1st ` R ) -> (GId ` g ) = (GId ` G ) )
10		drngi.4	. . . . . . . . . . 11 |- Z = (GId ` G )
11	9, 10	syl6eqr 2674	. . . . . . . . . 10 |- ( g = ( 1st ` R ) -> (GId ` g ) = Z )
12	11	sneqd 4189	. . . . . . . . 9 |- ( g = ( 1st ` R ) -> { (GId ` g ) } = { Z } )
13	8, 12	difeq12d 3729	. . . . . . . 8 |- ( g = ( 1st ` R ) -> ( ran g \ { (GId ` g ) } ) = ( X \ { Z } ) )
14	13	sqxpeqd 5141	. . . . . . 7 |- ( g = ( 1st ` R ) -> ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) = ( ( X \ { Z } ) X. ( X \ { Z } ) ) )
15	14	reseq2d 5396	. . . . . 6 |- ( g = ( 1st ` R ) -> ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) = ( h |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) )
16	15	eleq1d 2686	. . . . 5 |- ( g = ( 1st ` R ) -> ( ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp <-> ( h |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) )
17	2, 16	anbi12d 747	. . . 4 |- ( g = ( 1st ` R ) -> ( ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) <-> ( <. ( 1st ` R ) , h >. e. RingOps /\ ( h |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) ) )
18		opeq2 4403	. . . . . . 7 |- ( h = ( 2nd ` R ) -> <. ( 1st ` R ) , h >. = <. ( 1st ` R ) , ( 2nd ` R ) >. )
19	18	eleq1d 2686	. . . . . 6 |- ( h = ( 2nd ` R ) -> ( <. ( 1st ` R ) , h >. e. RingOps <-> <. ( 1st ` R ) , ( 2nd ` R ) >. e. RingOps ) )
20	19	anbi1d 741	. . . . 5 |- ( h = ( 2nd ` R ) -> ( ( <. ( 1st ` R ) , h >. e. RingOps /\ ( h |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) <-> ( <. ( 1st ` R ) , ( 2nd ` R ) >. e. RingOps /\ ( h |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) ) )
21		id 22	. . . . . . . . 9 |- ( h = ( 2nd ` R ) -> h = ( 2nd ` R ) )
22		drngi.2	. . . . . . . . 9 |- H = ( 2nd ` R )
23	21, 22	syl6reqr 2675	. . . . . . . 8 |- ( h = ( 2nd ` R ) -> H = h )
24	23	reseq1d 5395	. . . . . . 7 |- ( h = ( 2nd ` R ) -> ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) = ( h |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) )
25	24	eleq1d 2686	. . . . . 6 |- ( h = ( 2nd ` R ) -> ( ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp <-> ( h |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) )
26	25	anbi2d 740	. . . . 5 |- ( h = ( 2nd ` R ) -> ( ( <. ( 1st ` R ) , ( 2nd ` R ) >. e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) <-> ( <. ( 1st ` R ) , ( 2nd ` R ) >. e. RingOps /\ ( h |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) ) )
27	20, 26	bitr4d 271	. . . 4 |- ( h = ( 2nd ` R ) -> ( ( <. ( 1st ` R ) , h >. e. RingOps /\ ( h |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) <-> ( <. ( 1st ` R ) , ( 2nd ` R ) >. e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) ) )
28	17, 27	elopabi 7231	. . 3 |- ( R e. { <. g , h >. | ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) } -> ( <. ( 1st ` R ) , ( 2nd ` R ) >. e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) )
29		df-drngo 33748	. . 3 |- DivRingOps = { <. g , h >. | ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) }
30	28, 29	eleq2s 2719	. 2 |- ( R e. DivRingOps -> ( <. ( 1st ` R ) , ( 2nd ` R ) >. e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) )
31	29	relopabi 5245	. . . . 5 |- Rel DivRingOps
32		1st2nd 7214	. . . . 5 |- ( ( Rel DivRingOps /\ R e. DivRingOps ) -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
33	31, 32	mpan 706	. . . 4 |- ( R e. DivRingOps -> R = <. ( 1st ` R ) , ( 2nd ` R ) >. )
34	33	eleq1d 2686	. . 3 |- ( R e. DivRingOps -> ( R e. RingOps <-> <. ( 1st ` R ) , ( 2nd ` R ) >. e. RingOps ) )
35	34	anbi1d 741	. 2 |- ( R e. DivRingOps -> ( ( R e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) <-> ( <. ( 1st ` R ) , ( 2nd ` R ) >. e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) ) )
36	30, 35	mpbird 247	1 |- ( R e. DivRingOps -> ( R e. RingOps /\ ( H |` ( ( X \ { Z } ) X. ( X \ { Z } ) ) ) e. GrpOp ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   {csn 4177   <.cop 4183   {copab 4712   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-fv 5896  df-1st 7168  df-2nd 7169  df-drngo 33748

This theorem is referenced by:  dvrunz  33753  fldcrng  33803