Theorem isringd 16097

Description: Conditions that determine a ring.

Hypotheses

Ref	Expression
isringd.1	|- G = (1st` R)
isringd.2	|- H = (2nd` R)
isringd.3	|- (ph -> R e. (_V X. _V))
isringd.4	|- (ph -> G e. Abel)
isringd.5	|- (ph -> X = ran G)
isringd.6	|- (ph -> H:(X X. X)-->X)
isringd.7	|- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xHy)Hz) = (xH(yHz)))
isringd.8	|- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> (xH(yGz)) = ((xHy)G(xHz)))
isringd.9	|- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy)Hz) = ((xHz)G(yHz)))
isringd.10	|- (ph -> U e. X)
isringd.11	|- ((ph /\ y e. X) -> (yHU) = y)
isringd.12	|- ((ph /\ y e. X) -> (UHy) = y)

Assertion

Ref	Expression
isringd	|- (ph -> R e. Ring)

Distinct variable groups:   ph,x,y,z   x,G,y,z   x,H,y,z   x,U,y

Proof of Theorem isringd

Step	Hyp	Ref	Expression
1		1st2nd 5048	. . . 4 |- ((Rel (_V X. _V) /\ R e. (_V X. _V)) -> R = <.(1st` R), (2nd` R)>.)
2		relxp 4088	. . . 4 |- Rel (_V X. _V)
3		isringd.3	. . . 4 |- (ph -> R e. (_V X. _V))
4	1, 2, 3	sylancr 526	. . 3 |- (ph -> R = <.(1st` R), (2nd` R)>.)
5		isringd.1	. . . 4 |- G = (1st` R)
6		isringd.2	. . . 4 |- H = (2nd` R)
7	5, 6	opeq12i 3163	. . 3 |- <.G, H>. = <.(1st` R), (2nd` R)>.
8	4, 7	syl6eqr 1946	. 2 |- (ph -> R = <.G, H>.)
9		isringd.4	. . . 4 |- (ph -> G e. Abel)
10		isringd.6	. . . . 5 |- (ph -> H:(X X. X)-->X)
11		isringd.5	. . . . . . . 8 |- (ph -> X = ran G)
12		xpeq12 4020	. . . . . . . 8 |- ((X = ran G /\ X = ran G) -> (X X. X) = (ran G X. ran G))
13	11, 11, 12	syl11anc 524	. . . . . . 7 |- (ph -> (X X. X) = (ran G X. ran G))
14	13	feq2d 4557	. . . . . 6 |- (ph -> (H:(X X. X)-->X <-> H:(ran G X. ran G)-->X))
15		feq3 4553	. . . . . . 7 |- (X = ran G -> (H:(ran G X. ran G)-->X <-> H:(ran G X. ran G)-->ran G))
16	11, 15	syl 12	. . . . . 6 |- (ph -> (H:(ran G X. ran G)-->X <-> H:(ran G X. ran G)-->ran G))
17	14, 16	bitrd 587	. . . . 5 |- (ph -> (H:(X X. X)-->X <-> H:(ran G X. ran G)-->ran G))
18	10, 17	mpbid 212	. . . 4 |- (ph -> H:(ran G X. ran G)-->ran G)
19	11	eleq2d 1964	. . . . . . . . . . 11 |- (ph -> (x e. X <-> x e. ran G))
20	11	eleq2d 1964	. . . . . . . . . . 11 |- (ph -> (y e. X <-> y e. ran G))
21	11	eleq2d 1964	. . . . . . . . . . 11 |- (ph -> (z e. X <-> z e. ran G))
22	19, 20, 21	3anbi123d 1168	. . . . . . . . . 10 |- (ph -> ((x e. X /\ y e. X /\ z e. X) <-> (x e. ran G /\ y e. ran G /\ z e. ran G)))
23		isringd.7	. . . . . . . . . . . 12 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xHy)Hz) = (xH(yHz)))
24		isringd.8	. . . . . . . . . . . 12 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> (xH(yGz)) = ((xHy)G(xHz)))
25		isringd.9	. . . . . . . . . . . 12 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy)Hz) = ((xHz)G(yHz)))
26	23, 24, 25	3jca 1050	. . . . . . . . . . 11 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))))
27	26	ex 402	. . . . . . . . . 10 |- (ph -> ((x e. X /\ y e. X /\ z e. X) -> (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz)))))
28	22, 27	sylbird 222	. . . . . . . . 9 |- (ph -> ((x e. ran G /\ y e. ran G /\ z e. ran G) -> (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz)))))
29	28	3expd 1085	. . . . . . . 8 |- (ph -> (x e. ran G -> (y e. ran G -> (z e. ran G -> (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz)))))))
30	29	imp3a 388	. . . . . . 7 |- (ph -> ((x e. ran G /\ y e. ran G) -> (z e. ran G -> (((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))))))
31	30	r19.21adv 2181	. . . . . 6 |- (ph -> ((x e. ran G /\ y e. ran G) -> A.z e. ran G(((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz)))))
32	31	r19.21aivv 2183	. . . . 5 |- (ph -> A.x e. ran GA.y e. ran GA.z e. ran G(((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))))
33		isringd.10	. . . . . . 7 |- (ph -> U e. X)
34	33, 11	eleqtrd 1973	. . . . . 6 |- (ph -> U e. ran G)
35		isringd.11	. . . . . . . . . 10 |- ((ph /\ y e. X) -> (yHU) = y)
36		isringd.12	. . . . . . . . . 10 |- ((ph /\ y e. X) -> (UHy) = y)
37	35, 36	jca 310	. . . . . . . . 9 |- ((ph /\ y e. X) -> ((yHU) = y /\ (UHy) = y))
38	37	ex 402	. . . . . . . 8 |- (ph -> (y e. X -> ((yHU) = y /\ (UHy) = y)))
39	20, 38	sylbird 222	. . . . . . 7 |- (ph -> (y e. ran G -> ((yHU) = y /\ (UHy) = y)))
40	39	r19.21aiv 2175	. . . . . 6 |- (ph -> A.y e. ran G((yHU) = y /\ (UHy) = y))
41		opreq2 4890	. . . . . . . . . 10 |- (x = U -> (yHx) = (yHU))
42	41	eqeq1d 1892	. . . . . . . . 9 |- (x = U -> ((yHx) = y <-> (yHU) = y))
43		opreq1 4889	. . . . . . . . . 10 |- (x = U -> (xHy) = (UHy))
44	43	eqeq1d 1892	. . . . . . . . 9 |- (x = U -> ((xHy) = y <-> (UHy) = y))
45	42, 44	anbi12d 690	. . . . . . . 8 |- (x = U -> (((yHx) = y /\ (xHy) = y) <-> ((yHU) = y /\ (UHy) = y)))
46	45	ralbidv 2123	. . . . . . 7 |- (x = U -> (A.y e. ran G((yHx) = y /\ (xHy) = y) <-> A.y e. ran G((yHU) = y /\ (UHy) = y)))
47	46	rcla4ev 2381	. . . . . 6 |- ((U e. ran G /\ A.y e. ran G((yHU) = y /\ (UHy) = y)) -> E.x e. ran GA.y e. ran G((yHx) = y /\ (xHy) = y))
48	34, 40, 47	syl11anc 524	. . . . 5 |- (ph -> E.x e. ran GA.y e. ran G((yHx) = y /\ (xHy) = y))
49	32, 48	jca 310	. . . 4 |- (ph -> (A.x e. ran GA.y e. ran GA.z e. ran G(((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. ran GA.y e. ran G((yHx) = y /\ (xHy) = y)))
50	9, 18, 49	jca31 311	. . 3 |- (ph -> ((G e. Abel /\ H:(ran G X. ran G)-->ran G) /\ (A.x e. ran GA.y e. ran GA.z e. ran G(((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. ran GA.y e. ran G((yHx) = y /\ (xHy) = y))))
51		fvex 4689	. . . . 5 |- (2nd` R) e. _V
52	6, 51	eqeltri 1967	. . . 4 |- H e. _V
53		eqid 1884	. . . . 5 |- ran G = ran G
54	53	isring 9465	. . . 4 |- (H e. _V -> (<.G, H>. e. Ring <-> ((G e. Abel /\ H:(ran G X. ran G)-->ran G) /\ (A.x e. ran GA.y e. ran GA.z e. ran G(((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. ran GA.y e. ran G((yHx) = y /\ (xHy) = y)))))
55	52, 54	ax-mp 7	. . 3 |- (<.G, H>. e. Ring <-> ((G e. Abel /\ H:(ran G X. ran G)-->ran G) /\ (A.x e. ran GA.y e. ran GA.z e. ran G(((xHy)Hz) = (xH(yHz)) /\ (xH(yGz)) = ((xHy)G(xHz)) /\ ((xGy)Hz) = ((xHz)G(yHz))) /\ E.x e. ran GA.y e. ran G((yHx) = y /\ (xHy) = y))))
56	50, 55	sylibr 217	. 2 |- (ph -> <.G, H>. e. Ring)
57	8, 56	eqeltrd 1971	1 |- (ph -> R e. Ring)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292  <.cop 3046   X. cxp 3984  ran crn 3987  Rel wrel 3991  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Abelcabl 9407  Ringcring 9463

This theorem is referenced by:  iscringd 16147

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-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-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-ring 9464

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