Theorem isdivrngo 24063

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 4394	. . . . 5 |- ( G DivRingOps H <-> <. G , H >. e. DivRingOps )
2		df-drngo 24038	. . . . . . 7 |- DivRingOps = { <. x , y >. | ( <. x , y >. e. RingOps /\ ( y |` ( ( ran x \ { (GId ` x ) } ) X. ( ran x \ { (GId ` x ) } ) ) ) e. GrpOp ) }
3	2	relopabi 5066	. . . . . 6 |- Rel DivRingOps
4	3	brrelexi 4980	. . . . 5 |- ( G DivRingOps H -> G e. _V )
5	1, 4	sylbir 213	. . . 4 |- ( <. G , H >. e. DivRingOps -> G e. _V )
6	5	anim1i 568	. . 3 |- ( ( <. G , H >. e. DivRingOps /\ H e. A ) -> ( G e. _V /\ H e. A ) )
7	6	ancoms 453	. 2 |- ( ( H e. A /\ <. G , H >. e. DivRingOps ) -> ( G e. _V /\ H e. A ) )
8		rngoablo2 24054	. . . . 5 |- ( <. G , H >. e. RingOps -> G e. AbelOp )
9		elex 3080	. . . . 5 |- ( G e. AbelOp -> G e. _V )
10	8, 9	syl 16	. . . 4 |- ( <. G , H >. e. RingOps -> G e. _V )
11	10	ad2antrl 727	. . 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 457	. . 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 532	. 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 24038	. . . 4 |- DivRingOps = { <. g , h >. | ( <. g , h >. e. RingOps /\ ( h |` ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) ) e. GrpOp ) }
15	14	eleq2i 2529	. . 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 4160	. . . . . 6 |- ( g = G -> <. g , h >. = <. G , h >. )
17	16	eleq1d 2520	. . . . 5 |- ( g = G -> ( <. g , h >. e. RingOps <-> <. G , h >. e. RingOps ) )
18		rneq 5166	. . . . . . . . 9 |- ( g = G -> ran g = ran G )
19		fveq2 5792	. . . . . . . . . 10 |- ( g = G -> (GId ` g ) = (GId ` G ) )
20	19	sneqd 3990	. . . . . . . . 9 |- ( g = G -> { (GId ` g ) } = { (GId ` G ) } )
21	18, 20	difeq12d 3576	. . . . . . . 8 |- ( g = G -> ( ran g \ { (GId ` g ) } ) = ( ran G \ { (GId ` G ) } ) )
22	21, 21	xpeq12d 4966	. . . . . . 7 |- ( g = G -> ( ( ran g \ { (GId ` g ) } ) X. ( ran g \ { (GId ` g ) } ) ) = ( ( ran G \ { (GId ` G ) } ) X. ( ran G \ { (GId ` G ) } ) ) )
23	22	reseq2d 5211	. . . . . 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 2520	. . . . 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 710	. . . 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 4161	. . . . . 6 |- ( h = H -> <. G , h >. = <. G , H >. )
27	26	eleq1d 2520	. . . . 5 |- ( h = H -> ( <. G , h >. e. RingOps <-> <. G , H >. e. RingOps ) )
28		reseq1 5205	. . . . . 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 2520	. . . . 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 710	. . . 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 4708	. . 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 893	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 369   = wceq 1370   e. wcel 1758   _Vcvv 3071   \ cdif 3426   {csn 3978   <.cop 3984   class class class wbr 4393   {copab 4450   X. cxp 4939   ran crn 4942   |` cres 4943   ` cfv 5519   GrpOpcgr 23818  GIdcgi 23819   AbelOpcablo 23913   RingOpscrngo 24007   DivRingOpscdrng 24037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-1st 6680  df-2nd 6681  df-rngo 24008  df-drngo 24038

This theorem is referenced by:  zrdivrng  24064  isdrngo1  28903