Theorem isdivrng 10417

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

Assertion

Ref	Expression
isdivrng	|- (H e. A -> (<.G, H>. e. DivRing <-> (<.G, H>. e. Ring /\ (H |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp)))

Proof of Theorem isdivrng

Step	Hyp	Ref	Expression
1		df-br 3339	. . . . 5 |- (GDivRingH <-> <.G, H>. e. DivRing)
2		relopab 4104	. . . . . . 7 |- Rel {<.x, y>. | (<.x, y>. e. Ring /\ (y |` ((ran x \ {(Id` x)}) X. (ran x \ {(Id` x)}))) e. Grp)}
3		df-drng 9492	. . . . . . . 8 |- DivRing = {<.x, y>. | (<.x, y>. e. Ring /\ (y |` ((ran x \ {(Id` x)}) X. (ran x \ {(Id` x)}))) e. Grp)}
4	3	releqi 4072	. . . . . . 7 |- (Rel DivRing <-> Rel {<.x, y>. | (<.x, y>. e. Ring /\ (y |` ((ran x \ {(Id` x)}) X. (ran x \ {(Id` x)}))) e. Grp)})
5	2, 4	mpbir 207	. . . . . 6 |- Rel DivRing
6	5	brrelexi 4029	. . . . 5 |- (GDivRingH -> G e. _V)
7	1, 6	sylbir 218	. . . 4 |- (<.G, H>. e. DivRing -> G e. _V)
8	7	anim1i 361	. . 3 |- ((<.G, H>. e. DivRing /\ H e. A) -> (G e. _V /\ H e. A))
9	8	ancoms 484	. 2 |- ((H e. A /\ <.G, H>. e. DivRing) -> (G e. _V /\ H e. A))
10		fora 10408	. . . . 5 |- (<.G, H>. e. Ring -> G e. Abel)
11		elisset 2299	. . . . 5 |- (G e. Abel -> G e. _V)
12	10, 11	syl 12	. . . 4 |- (<.G, H>. e. Ring -> G e. _V)
13	12	ad2antrl 442	. . 3 |- ((H e. A /\ (<.G, H>. e. Ring /\ (H |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp)) -> G e. _V)
14		simpl 346	. . 3 |- ((H e. A /\ (<.G, H>. e. Ring /\ (H |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp)) -> H e. A)
15	13, 14	jca 310	. 2 |- ((H e. A /\ (<.G, H>. e. Ring /\ (H |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp)) -> (G e. _V /\ H e. A))
16		opeq1 3158	. . . . . 6 |- (g = G -> <.g, h>. = <.G, h>.)
17	16	eleq1d 1963	. . . . 5 |- (g = G -> (<.g, h>. e. Ring <-> <.G, h>. e. Ring))
18		rneq 4186	. . . . . . . . 9 |- (g = G -> ran g = ran G)
19		fveq2 4681	. . . . . . . . . 10 |- (g = G -> (Id` g) = (Id` G))
20	19	sneqd 3056	. . . . . . . . 9 |- (g = G -> {(Id` g)} = {(Id` G)})
21		difeq12 2721	. . . . . . . . 9 |- ((ran g = ran G /\ {(Id` g)} = {(Id` G)}) -> (ran g \ {(Id` g)}) = (ran G \ {(Id` G)}))
22	18, 20, 21	syl11anc 524	. . . . . . . 8 |- (g = G -> (ran g \ {(Id` g)}) = (ran G \ {(Id` G)}))
23		xpeq12 4020	. . . . . . . 8 |- (((ran g \ {(Id` g)}) = (ran G \ {(Id` G)}) /\ (ran g \ {(Id` g)}) = (ran G \ {(Id` G)})) -> ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)})) = ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)})))
24	22, 22, 23	syl11anc 524	. . . . . . 7 |- (g = G -> ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)})) = ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)})))
25		reseq2 4219	. . . . . . 7 |- (((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)})) = ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)})) -> (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) = (h |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))))
26	24, 25	syl 12	. . . . . 6 |- (g = G -> (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) = (h |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))))
27	26	eleq1d 1963	. . . . 5 |- (g = G -> ((h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp <-> (h |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp))
28	17, 27	anbi12d 690	. . . 4 |- (g = G -> ((<.g, h>. e. Ring /\ (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp) <-> (<.G, h>. e. Ring /\ (h |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp)))
29		opeq2 3159	. . . . . 6 |- (h = H -> <.G, h>. = <.G, H>.)
30	29	eleq1d 1963	. . . . 5 |- (h = H -> (<.G, h>. e. Ring <-> <.G, H>. e. Ring))
31		reseq1 4218	. . . . . 6 |- (h = H -> (h |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) = (H |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))))
32	31	eleq1d 1963	. . . . 5 |- (h = H -> ((h |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp <-> (H |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp))
33	30, 32	anbi12d 690	. . . 4 |- (h = H -> ((<.G, h>. e. Ring /\ (h |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp) <-> (<.G, H>. e. Ring /\ (H |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp)))
34	28, 33	opelopabg 3567	. . 3 |- ((G e. _V /\ H e. A) -> (<.G, H>. e. {<.g, h>. | (<.g, h>. e. Ring /\ (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp)} <-> (<.G, H>. e. Ring /\ (H |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp)))
35		df-drng 9492	. . . 4 |- DivRing = {<.g, h>. | (<.g, h>. e. Ring /\ (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp)}
36	35	eleq2i 1961	. . 3 |- (<.G, H>. e. DivRing <-> <.G, H>. e. {<.g, h>. | (<.g, h>. e. Ring /\ (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp)})
37	34, 36	syl5bb 591	. 2 |- ((G e. _V /\ H e. A) -> (<.G, H>. e. DivRing <-> (<.G, H>. e. Ring /\ (H |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp)))
38	9, 15, 37	pm5.21nd 744	1 |- (H e. A -> (<.G, H>. e. DivRing <-> (<.G, H>. e. Ring /\ (H |` ((ran G \ {(Id` G)}) X. (ran G \ {(Id` G)}))) e. Grp)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   \ cdif 2590  {csn 3044  <.cop 3046   class class class wbr 3338  {copab 3395   X. cxp 3984  ran crn 3987   |` cres 3988  Rel wrel 3991  ` cfv 3998  Grpcgr 9311  Idcgi 9312  Abelcabl 9407  Ringcring 9463  DivRingcdrng 9491

This theorem is referenced by:  zrdivrng 10418  isdivrng1 16109

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-ring 9464  df-drng 9492

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