Theorem isdivrngo 33749

Description: The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
isdivrngo	|- ( H e. A -> ( <. G , H >. e. DivRingOps <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) ) )

Proof of Theorem isdivrngo

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

Step	Hyp	Ref	Expression
1		df-br 4654	. . . . 5 |- ( G DivRingOps H <-> <. G , H >. e. DivRingOps )
2		df-drngo 33748	. . . . . . 7 |- DivRingOps = { <. x , y >. | ( <. x , y >. e. RingOps /\ ( y |` ( ( ran x \ { (GId ` x ) } ) X. ( ran x \ { (GId ` x ) } ) ) ) e. GrpOp ) }
3	2	relopabi 5245	. . . . . 6 |- Rel DivRingOps
4	3	brrelexi 5158	. . . . 5 |- ( G DivRingOps H -> G e. _V )
5	1, 4	sylbir 225	. . . 4 |- ( <. G , H >. e. DivRingOps -> G e. _V )
6	5	anim1i 592	. . 3 |- ( ( <. G , H >. e. DivRingOps /\ H e. A ) -> ( G e. _V /\ H e. A ) )
7	6	ancoms 469	. 2 |- ( ( H e. A /\ <. G , H >. e. DivRingOps ) -> ( G e. _V /\ H e. A ) )
8		rngoablo2 33708	. . . . 5 |- ( <. G , H >. e. RingOps -> G e. AbelOp )
9		elex 3212	. . . . 5 |- ( G e. AbelOp -> G e. _V )
10	8, 9	syl 17	. . . 4 |- ( <. G , H >. e. RingOps -> G e. _V )
11	10	ad2antrl 764	. . 3 |- ( ( H e. A /\ ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) ) -> G e. _V )
12		simpl 473	. . 3 |- ( ( H e. A /\ ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) ) -> H e. A )
13	11, 12	jca 554	. 2 |- ( ( H e. A /\ ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) ) -> ( G e. _V /\ H e. A ) )
14		df-drngo 33748	. . . 4 |- DivRingOps = { <. g , h >. | ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) }
15	14	eleq2i 2693	. . 3 |- ( <. G , H >. e. DivRingOps <-> <. G , H >. e. { <. g , h >. | ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) } )
16		opeq1 4402	. . . . . 6 |- ( g = G -> <. g , h >. = <. G , h >. )
17	16	eleq1d 2686	. . . . 5 |- ( g = G -> ( <. g , h >. e. RingOps <-> <. G , h >. e. RingOps ) )
18		rneq 5351	. . . . . . . . 9 |- ( g = G -> ran g = ran G )
19		fveq2 6191	. . . . . . . . . 10 |- ( g = G -> (GId ` g ) = (GId ` G ) )
20	19	sneqd 4189	. . . . . . . . 9 |- ( g = G -> { (GId ` g ) } = { (GId ` G ) } )
21	18, 20	difeq12d 3729	. . . . . . . 8 |- ( g = G -> ( ran g \ { (GId ` g ) } ) = ( ran G \ { (GId ` G ) } ) )
22	21	sqxpeqd 5141	. . . . . . 7 |- ( g = G -> ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) = ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) )
23	22	reseq2d 5396	. . . . . 6 |- ( g = G -> ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) = ( h |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) )
24	23	eleq1d 2686	. . . . 5 |- ( g = G -> ( ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp <-> ( h |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) )
25	17, 24	anbi12d 747	. . . 4 |- ( g = G -> ( ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) <-> ( <. G , h >. e. RingOps /\ ( h |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) ) )
26		opeq2 4403	. . . . . 6 |- ( h = H -> <. G , h >. = <. G , H >. )
27	26	eleq1d 2686	. . . . 5 |- ( h = H -> ( <. G , h >. e. RingOps <-> <. G , H >. e. RingOps ) )
28		reseq1 5390	. . . . . 6 |- ( h = H -> ( h |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) = ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) )
29	28	eleq1d 2686	. . . . 5 |- ( h = H -> ( ( h |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp <-> ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) )
30	27, 29	anbi12d 747	. . . 4 |- ( h = H -> ( ( <. G , h >. e. RingOps /\ ( h |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) ) )
31	25, 30	opelopabg 4993	. . 3 |- ( ( G e. _V /\ H e. A ) -> ( <. G , H >. e. { <. g , h >. | ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) } <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) ) )
32	15, 31	syl5bb 272	. 2 |- ( ( G e. _V /\ H e. A ) -> ( <. G , H >. e. DivRingOps <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) ) )
33	7, 13, 32	pm5.21nd 941	1 |- ( H e. A -> ( <. G , H >. e. DivRingOps <-> ( <. G , H >. e. RingOps /\ ( H |` ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) ) e. GrpOp ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   {csn 4177   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112   ran crn 5115   |` cres 5116   ` cfv 5888   GrpOpcgr 27343  GIdcgi 27344   AbelOpcablo 27398   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:  zrdivrng  33752  isdrngo1  33755