Theorem isdivrngo 25631

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 4440	. . . . 5 |- ( G DivRingOps H <-> <. G , H >. e. DivRingOps )
2		df-drngo 25606	. . . . . . 7 |- DivRingOps = { <. x , y >. | ( <. x , y >. e. RingOps /\ ( y |` ( ( ran x \ { (GId ` x ) } ) X. ( ran x \ { (GId ` x ) } ) ) ) e. GrpOp ) }
3	2	relopabi 5116	. . . . . 6 |- Rel DivRingOps
4	3	brrelexi 5029	. . . . 5 |- ( G DivRingOps H -> G e. _V )
5	1, 4	sylbir 213	. . . 4 |- ( <. G , H >. e. DivRingOps -> G e. _V )
6	5	anim1i 566	. . 3 |- ( ( <. G , H >. e. DivRingOps /\ H e. A ) -> ( G e. _V /\ H e. A ) )
7	6	ancoms 451	. 2 |- ( ( H e. A /\ <. G , H >. e. DivRingOps ) -> ( G e. _V /\ H e. A ) )
8		rngoablo2 25622	. . . . 5 |- ( <. G , H >. e. RingOps -> G e. AbelOp )
9		elex 3115	. . . . 5 |- ( G e. AbelOp -> G e. _V )
10	8, 9	syl 16	. . . 4 |- ( <. G , H >. e. RingOps -> G e. _V )
11	10	ad2antrl 725	. . 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 455	. . 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 530	. 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 25606	. . . 4 |- DivRingOps = { <. g , h >. | ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) }
15	14	eleq2i 2532	. . 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 4203	. . . . . 6 |- ( g = G -> <. g , h >. = <. G , h >. )
17	16	eleq1d 2523	. . . . 5 |- ( g = G -> ( <. g , h >. e. RingOps <-> <. G , h >. e. RingOps ) )
18		rneq 5217	. . . . . . . . 9 |- ( g = G -> ran g = ran G )
19		fveq2 5848	. . . . . . . . . 10 |- ( g = G -> (GId ` g ) = (GId ` G ) )
20	19	sneqd 4028	. . . . . . . . 9 |- ( g = G -> { (GId ` g ) } = { (GId ` G ) } )
21	18, 20	difeq12d 3609	. . . . . . . 8 |- ( g = G -> ( ran g \ { (GId ` g ) } ) = ( ran G \ { (GId ` G ) } ) )
22	21	sqxpeqd 5014	. . . . . . 7 |- ( g = G -> ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) = ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) )
23	22	reseq2d 5262	. . . . . 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 2523	. . . . 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 708	. . . 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 4204	. . . . . 6 |- ( h = H -> <. G , h >. = <. G , H >. )
27	26	eleq1d 2523	. . . . 5 |- ( h = H -> ( <. G , h >. e. RingOps <-> <. G , H >. e. RingOps ) )
28		reseq1 5256	. . . . . 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 2523	. . . . 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 708	. . . 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 4754	. . 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 257	. 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 898	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 184   /\ wa 367   = wceq 1398   e. wcel 1823   _Vcvv 3106   \ cdif 3458   {csn 4016   <.cop 4022   class class class wbr 4439   {copab 4496   X. cxp 4986   ran crn 4989   |` cres 4990   ` cfv 5570   GrpOpcgr 25386  GIdcgi 25387   AbelOpcablo 25481   RingOpscrngo 25575   DivRingOpscdrng 25605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-1st 6773  df-2nd 6774  df-rngo 25576  df-drngo 25606

This theorem is referenced by:  zrdivrng  25632  isdrngo1  30599