Theorem drngi 9493

Description: The properties of a division ring.

Hypotheses

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

Assertion

Ref	Expression
drngi	|- (R e. DivRing -> (R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp))

Proof of Theorem drngi

Step	Hyp	Ref	Expression
1		df-drng 9492	. . . 4 |- DivRing = {<.g, h>. | (<.g, h>. e. Ring /\ (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp)}
2	1	eleq2i 1961	. . 3 |- (R e. DivRing <-> R e. {<.g, h>. | (<.g, h>. e. Ring /\ (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp)})
3		opeq1 3158	. . . . . 6 |- (g = (1st` R) -> <.g, h>. = <.(1st` R), h>.)
4	3	eleq1d 1963	. . . . 5 |- (g = (1st` R) -> (<.g, h>. e. Ring <-> <.(1st` R), h>. e. Ring))
5		id 73	. . . . . . . . . . . . 13 |- (g = (1st` R) -> g = (1st` R))
6		drngi.1	. . . . . . . . . . . . 13 |- G = (1st` R)
7	5, 6	syl6eqr 1946	. . . . . . . . . . . 12 |- (g = (1st` R) -> g = G)
8	7	rneqd 4188	. . . . . . . . . . 11 |- (g = (1st` R) -> ran g = ran G)
9		drngi.3	. . . . . . . . . . 11 |- X = ran G
10	8, 9	syl6eqr 1946	. . . . . . . . . 10 |- (g = (1st` R) -> ran g = X)
11	10	difeq1d 2725	. . . . . . . . 9 |- (g = (1st` R) -> (ran g \ {(Id` g)}) = (X \ {(Id` g)}))
12	7	fveq2d 4685	. . . . . . . . . . . 12 |- (g = (1st` R) -> (Id` g) = (Id` G))
13		drngi.4	. . . . . . . . . . . 12 |- Z = (Id` G)
14	12, 13	syl6eqr 1946	. . . . . . . . . . 11 |- (g = (1st` R) -> (Id` g) = Z)
15	14	sneqd 3056	. . . . . . . . . 10 |- (g = (1st` R) -> {(Id` g)} = {Z})
16	15	difeq2d 2726	. . . . . . . . 9 |- (g = (1st` R) -> (X \ {(Id` g)}) = (X \ {Z}))
17	11, 16	eqtrd 1925	. . . . . . . 8 |- (g = (1st` R) -> (ran g \ {(Id` g)}) = (X \ {Z}))
18		xpeq2 4017	. . . . . . . . 9 |- ((ran g \ {(Id` g)}) = (X \ {Z}) -> ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)})) = ((ran g \ {(Id` g)}) X. (X \ {Z})))
19		xpeq1 4016	. . . . . . . . 9 |- ((ran g \ {(Id` g)}) = (X \ {Z}) -> ((ran g \ {(Id` g)}) X. (X \ {Z})) = ((X \ {Z}) X. (X \ {Z})))
20	18, 19	eqtrd 1925	. . . . . . . 8 |- ((ran g \ {(Id` g)}) = (X \ {Z}) -> ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)})) = ((X \ {Z}) X. (X \ {Z})))
21	17, 20	syl 12	. . . . . . 7 |- (g = (1st` R) -> ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)})) = ((X \ {Z}) X. (X \ {Z})))
22		reseq2 4219	. . . . . . 7 |- (((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)})) = ((X \ {Z}) X. (X \ {Z})) -> (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) = (h |` ((X \ {Z}) X. (X \ {Z}))))
23	21, 22	syl 12	. . . . . 6 |- (g = (1st` R) -> (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) = (h |` ((X \ {Z}) X. (X \ {Z}))))
24	23	eleq1d 1963	. . . . 5 |- (g = (1st` R) -> ((h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp <-> (h |` ((X \ {Z}) X. (X \ {Z}))) e. Grp))
25	4, 24	anbi12d 690	. . . 4 |- (g = (1st` R) -> ((<.g, h>. e. Ring /\ (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp) <-> (<.(1st` R), h>. e. Ring /\ (h |` ((X \ {Z}) X. (X \ {Z}))) e. Grp)))
26		opeq2 3159	. . . . . . 7 |- (h = (2nd` R) -> <.(1st` R), h>. = <.(1st` R), (2nd` R)>.)
27	26	eleq1d 1963	. . . . . 6 |- (h = (2nd` R) -> (<.(1st` R), h>. e. Ring <-> <.(1st` R), (2nd` R)>. e. Ring))
28	27	anbi1d 679	. . . . 5 |- (h = (2nd` R) -> ((<.(1st` R), h>. e. Ring /\ (h |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) <-> (<.(1st` R), (2nd` R)>. e. Ring /\ (h |` ((X \ {Z}) X. (X \ {Z}))) e. Grp)))
29		id 73	. . . . . . . . 9 |- (h = (2nd` R) -> h = (2nd` R))
30		drngi.2	. . . . . . . . 9 |- H = (2nd` R)
31	29, 30	syl6reqr 1947	. . . . . . . 8 |- (h = (2nd` R) -> H = h)
32		reseq1 4218	. . . . . . . 8 |- (H = h -> (H |` ((X \ {Z}) X. (X \ {Z}))) = (h |` ((X \ {Z}) X. (X \ {Z}))))
33	31, 32	syl 12	. . . . . . 7 |- (h = (2nd` R) -> (H |` ((X \ {Z}) X. (X \ {Z}))) = (h |` ((X \ {Z}) X. (X \ {Z}))))
34	33	eleq1d 1963	. . . . . 6 |- (h = (2nd` R) -> ((H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp <-> (h |` ((X \ {Z}) X. (X \ {Z}))) e. Grp))
35	34	anbi2d 678	. . . . 5 |- (h = (2nd` R) -> ((<.(1st` R), (2nd` R)>. e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) <-> (<.(1st` R), (2nd` R)>. e. Ring /\ (h |` ((X \ {Z}) X. (X \ {Z}))) e. Grp)))
36	28, 35	bitr4d 590	. . . 4 |- (h = (2nd` R) -> ((<.(1st` R), h>. e. Ring /\ (h |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) <-> (<.(1st` R), (2nd` R)>. e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp)))
37	25, 36	elopabi 5059	. . 3 |- (R e. {<.g, h>. | (<.g, h>. e. Ring /\ (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp)} -> (<.(1st` R), (2nd` R)>. e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp))
38	2, 37	sylbi 216	. 2 |- (R e. DivRing -> (<.(1st` R), (2nd` R)>. e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp))
39		relopab 4104	. . . . . 6 |- Rel {<.g, h>. | (<.g, h>. e. Ring /\ (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp)}
40	1	releqi 4072	. . . . . 6 |- (Rel DivRing <-> Rel {<.g, h>. | (<.g, h>. e. Ring /\ (h |` ((ran g \ {(Id` g)}) X. (ran g \ {(Id` g)}))) e. Grp)})
41	39, 40	mpbir 207	. . . . 5 |- Rel DivRing
42		1st2nd 5048	. . . . 5 |- ((Rel DivRing /\ R e. DivRing) -> R = <.(1st` R), (2nd` R)>.)
43	41, 42	mpan 759	. . . 4 |- (R e. DivRing -> R = <.(1st` R), (2nd` R)>.)
44	43	eleq1d 1963	. . 3 |- (R e. DivRing -> (R e. Ring <-> <.(1st` R), (2nd` R)>. e. Ring))
45	44	anbi1d 679	. 2 |- (R e. DivRing -> ((R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp) <-> (<.(1st` R), (2nd` R)>. e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp)))
46	38, 45	mpbird 213	1 |- (R e. DivRing -> (R e. Ring /\ (H |` ((X \ {Z}) X. (X \ {Z}))) e. Grp))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590  {csn 3044  <.cop 3046  {copab 3395   X. cxp 3984  ran crn 3987   |` cres 3988  Rel wrel 3991  ` cfv 3998  1stc1st 5018  2ndc2nd 5019  Grpcgr 9311  Idcgi 9312  Ringcring 9463  DivRingcdrng 9491

This theorem is referenced by:  dvrunz 10419  fldi 14776  fldcrng 16152

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-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-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-fv 4014  df-1st 5020  df-2nd 5021  df-drng 9492

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