Theorem iscringd 16147

Description: Conditions that determine a commutative ring.

Hypotheses

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

Assertion

Ref	Expression
iscringd	|- (ph -> R e. CRing)

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

Proof of Theorem iscringd

Step	Hyp	Ref	Expression
1		iscringd.1	. . 3 |- G = (1st` R)
2		iscringd.2	. . 3 |- H = (2nd` R)
3		eqid 1884	. . 3 |- ran G = ran G
4	1, 2, 3	iscring2 16146	. 2 |- (R e. CRing <-> (R e. Ring /\ A.x e. ran GA.y e. ran G(xHy) = (yHx)))
5		iscringd.3	. . 3 |- (ph -> R e. (_V X. _V))
6		iscringd.4	. . 3 |- (ph -> G e. Abel)
7		iscringd.5	. . 3 |- (ph -> X = ran G)
8		iscringd.6	. . 3 |- (ph -> H:(X X. X)-->X)
9		iscringd.7	. . 3 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xHy)Hz) = (xH(yHz)))
10		iscringd.8	. . 3 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> (xH(yGz)) = ((xHy)G(xHz)))
11		eleq1 1957	. . . . . . . . 9 |- (w = z -> (w e. X <-> z e. X))
12	11	3anbi1d 1172	. . . . . . . 8 |- (w = z -> ((w e. X /\ x e. X /\ y e. X) <-> (z e. X /\ x e. X /\ y e. X)))
13	12	anbi2d 678	. . . . . . 7 |- (w = z -> ((ph /\ (w e. X /\ x e. X /\ y e. X)) <-> (ph /\ (z e. X /\ x e. X /\ y e. X))))
14		opreq1 4889	. . . . . . . 8 |- (w = z -> (wH(xGy)) = (zH(xGy)))
15		opreq1 4889	. . . . . . . . 9 |- (w = z -> (wHx) = (zHx))
16		opreq1 4889	. . . . . . . . 9 |- (w = z -> (wHy) = (zHy))
17	15, 16	opreq12d 4900	. . . . . . . 8 |- (w = z -> ((wHx)G(wHy)) = ((zHx)G(zHy)))
18	14, 17	eqeq12d 1899	. . . . . . 7 |- (w = z -> ((wH(xGy)) = ((wHx)G(wHy)) <-> (zH(xGy)) = ((zHx)G(zHy))))
19	13, 18	imbi12d 688	. . . . . 6 |- (w = z -> (((ph /\ (w e. X /\ x e. X /\ y e. X)) -> (wH(xGy)) = ((wHx)G(wHy))) <-> ((ph /\ (z e. X /\ x e. X /\ y e. X)) -> (zH(xGy)) = ((zHx)G(zHy)))))
20		eleq1 1957	. . . . . . . . . 10 |- (z = y -> (z e. X <-> y e. X))
21	20	3anbi3d 1174	. . . . . . . . 9 |- (z = y -> ((w e. X /\ x e. X /\ z e. X) <-> (w e. X /\ x e. X /\ y e. X)))
22	21	anbi2d 678	. . . . . . . 8 |- (z = y -> ((ph /\ (w e. X /\ x e. X /\ z e. X)) <-> (ph /\ (w e. X /\ x e. X /\ y e. X))))
23		opreq2 4890	. . . . . . . . . 10 |- (z = y -> (xGz) = (xGy))
24	23	opreq2d 4898	. . . . . . . . 9 |- (z = y -> (wH(xGz)) = (wH(xGy)))
25		opreq2 4890	. . . . . . . . . 10 |- (z = y -> (wHz) = (wHy))
26	25	opreq2d 4898	. . . . . . . . 9 |- (z = y -> ((wHx)G(wHz)) = ((wHx)G(wHy)))
27	24, 26	eqeq12d 1899	. . . . . . . 8 |- (z = y -> ((wH(xGz)) = ((wHx)G(wHz)) <-> (wH(xGy)) = ((wHx)G(wHy))))
28	22, 27	imbi12d 688	. . . . . . 7 |- (z = y -> (((ph /\ (w e. X /\ x e. X /\ z e. X)) -> (wH(xGz)) = ((wHx)G(wHz))) <-> ((ph /\ (w e. X /\ x e. X /\ y e. X)) -> (wH(xGy)) = ((wHx)G(wHy)))))
29		eleq1 1957	. . . . . . . . . . 11 |- (y = x -> (y e. X <-> x e. X))
30	29	3anbi2d 1173	. . . . . . . . . 10 |- (y = x -> ((w e. X /\ y e. X /\ z e. X) <-> (w e. X /\ x e. X /\ z e. X)))
31	30	anbi2d 678	. . . . . . . . 9 |- (y = x -> ((ph /\ (w e. X /\ y e. X /\ z e. X)) <-> (ph /\ (w e. X /\ x e. X /\ z e. X))))
32		opreq1 4889	. . . . . . . . . . 11 |- (y = x -> (yGz) = (xGz))
33	32	opreq2d 4898	. . . . . . . . . 10 |- (y = x -> (wH(yGz)) = (wH(xGz)))
34		opreq2 4890	. . . . . . . . . . 11 |- (y = x -> (wHy) = (wHx))
35	34	opreq1d 4897	. . . . . . . . . 10 |- (y = x -> ((wHy)G(wHz)) = ((wHx)G(wHz)))
36	33, 35	eqeq12d 1899	. . . . . . . . 9 |- (y = x -> ((wH(yGz)) = ((wHy)G(wHz)) <-> (wH(xGz)) = ((wHx)G(wHz))))
37	31, 36	imbi12d 688	. . . . . . . 8 |- (y = x -> (((ph /\ (w e. X /\ y e. X /\ z e. X)) -> (wH(yGz)) = ((wHy)G(wHz))) <-> ((ph /\ (w e. X /\ x e. X /\ z e. X)) -> (wH(xGz)) = ((wHx)G(wHz)))))
38		eleq1 1957	. . . . . . . . . . . 12 |- (x = w -> (x e. X <-> w e. X))
39	38	3anbi1d 1172	. . . . . . . . . . 11 |- (x = w -> ((x e. X /\ y e. X /\ z e. X) <-> (w e. X /\ y e. X /\ z e. X)))
40	39	anbi2d 678	. . . . . . . . . 10 |- (x = w -> ((ph /\ (x e. X /\ y e. X /\ z e. X)) <-> (ph /\ (w e. X /\ y e. X /\ z e. X))))
41		opreq1 4889	. . . . . . . . . . 11 |- (x = w -> (xH(yGz)) = (wH(yGz)))
42		opreq1 4889	. . . . . . . . . . . 12 |- (x = w -> (xHy) = (wHy))
43		opreq1 4889	. . . . . . . . . . . 12 |- (x = w -> (xHz) = (wHz))
44	42, 43	opreq12d 4900	. . . . . . . . . . 11 |- (x = w -> ((xHy)G(xHz)) = ((wHy)G(wHz)))
45	41, 44	eqeq12d 1899	. . . . . . . . . 10 |- (x = w -> ((xH(yGz)) = ((xHy)G(xHz)) <-> (wH(yGz)) = ((wHy)G(wHz))))
46	40, 45	imbi12d 688	. . . . . . . . 9 |- (x = w -> (((ph /\ (x e. X /\ y e. X /\ z e. X)) -> (xH(yGz)) = ((xHy)G(xHz))) <-> ((ph /\ (w e. X /\ y e. X /\ z e. X)) -> (wH(yGz)) = ((wHy)G(wHz)))))
47	46, 10	chvarv 1712	. . . . . . . 8 |- ((ph /\ (w e. X /\ y e. X /\ z e. X)) -> (wH(yGz)) = ((wHy)G(wHz)))
48	37, 47	chvarv 1712	. . . . . . 7 |- ((ph /\ (w e. X /\ x e. X /\ z e. X)) -> (wH(xGz)) = ((wHx)G(wHz)))
49	28, 48	chvarv 1712	. . . . . 6 |- ((ph /\ (w e. X /\ x e. X /\ y e. X)) -> (wH(xGy)) = ((wHx)G(wHy)))
50	19, 49	chvarv 1712	. . . . 5 |- ((ph /\ (z e. X /\ x e. X /\ y e. X)) -> (zH(xGy)) = ((zHx)G(zHy)))
51		3anrot 863	. . . . 5 |- ((z e. X /\ x e. X /\ y e. X) <-> (x e. X /\ y e. X /\ z e. X))
52	50, 51	sylan2br 502	. . . 4 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> (zH(xGy)) = ((zHx)G(zHy)))
53	7	eleq2d 1964	. . . . . . . . . 10 |- (ph -> (x e. X <-> x e. ran G))
54	7	eleq2d 1964	. . . . . . . . . 10 |- (ph -> (y e. X <-> y e. ran G))
55	53, 54	anbi12d 690	. . . . . . . . 9 |- (ph -> ((x e. X /\ y e. X) <-> (x e. ran G /\ y e. ran G)))
56	55	biimpa 460	. . . . . . . 8 |- ((ph /\ (x e. X /\ y e. X)) -> (x e. ran G /\ y e. ran G))
57	3	grpcl 9324	. . . . . . . . . . 11 |- ((G e. Grp /\ x e. ran G /\ y e. ran G) -> (xGy) e. ran G)
58	57	3expb 1068	. . . . . . . . . 10 |- ((G e. Grp /\ (x e. ran G /\ y e. ran G)) -> (xGy) e. ran G)
59		ablgrp 9410	. . . . . . . . . . 11 |- (G e. Abel -> G e. Grp)
60	6, 59	syl 12	. . . . . . . . . 10 |- (ph -> G e. Grp)
61	58, 60	sylan 497	. . . . . . . . 9 |- ((ph /\ (x e. ran G /\ y e. ran G)) -> (xGy) e. ran G)
62	7	eleq2d 1964	. . . . . . . . . 10 |- (ph -> ((xGy) e. X <-> (xGy) e. ran G))
63	62	adantr 425	. . . . . . . . 9 |- ((ph /\ (x e. ran G /\ y e. ran G)) -> ((xGy) e. X <-> (xGy) e. ran G))
64	61, 63	mpbird 213	. . . . . . . 8 |- ((ph /\ (x e. ran G /\ y e. ran G)) -> (xGy) e. X)
65	56, 64	syldan 516	. . . . . . 7 |- ((ph /\ (x e. X /\ y e. X)) -> (xGy) e. X)
66	65	3adantr3 1037	. . . . . 6 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> (xGy) e. X)
67		simpr3 884	. . . . . 6 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> z e. X)
68	66, 67	jca 310	. . . . 5 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy) e. X /\ z e. X))
69		oprex 4907	. . . . . 6 |- (xGy) e. _V
70		eleq1 1957	. . . . . . . . 9 |- (w = (xGy) -> (w e. X <-> (xGy) e. X))
71	70	anbi1d 679	. . . . . . . 8 |- (w = (xGy) -> ((w e. X /\ z e. X) <-> ((xGy) e. X /\ z e. X)))
72	71	anbi2d 678	. . . . . . 7 |- (w = (xGy) -> ((ph /\ (w e. X /\ z e. X)) <-> (ph /\ ((xGy) e. X /\ z e. X))))
73		opreq1 4889	. . . . . . . 8 |- (w = (xGy) -> (wHz) = ((xGy)Hz))
74		opreq2 4890	. . . . . . . 8 |- (w = (xGy) -> (zHw) = (zH(xGy)))
75	73, 74	eqeq12d 1899	. . . . . . 7 |- (w = (xGy) -> ((wHz) = (zHw) <-> ((xGy)Hz) = (zH(xGy))))
76	72, 75	imbi12d 688	. . . . . 6 |- (w = (xGy) -> (((ph /\ (w e. X /\ z e. X)) -> (wHz) = (zHw)) <-> ((ph /\ ((xGy) e. X /\ z e. X)) -> ((xGy)Hz) = (zH(xGy)))))
77		eleq1 1957	. . . . . . . . . 10 |- (y = z -> (y e. X <-> z e. X))
78	77	anbi2d 678	. . . . . . . . 9 |- (y = z -> ((w e. X /\ y e. X) <-> (w e. X /\ z e. X)))
79	78	anbi2d 678	. . . . . . . 8 |- (y = z -> ((ph /\ (w e. X /\ y e. X)) <-> (ph /\ (w e. X /\ z e. X))))
80		opreq2 4890	. . . . . . . . 9 |- (y = z -> (wHy) = (wHz))
81		opreq1 4889	. . . . . . . . 9 |- (y = z -> (yHw) = (zHw))
82	80, 81	eqeq12d 1899	. . . . . . . 8 |- (y = z -> ((wHy) = (yHw) <-> (wHz) = (zHw)))
83	79, 82	imbi12d 688	. . . . . . 7 |- (y = z -> (((ph /\ (w e. X /\ y e. X)) -> (wHy) = (yHw)) <-> ((ph /\ (w e. X /\ z e. X)) -> (wHz) = (zHw))))
84	38	anbi1d 679	. . . . . . . . . 10 |- (x = w -> ((x e. X /\ y e. X) <-> (w e. X /\ y e. X)))
85	84	anbi2d 678	. . . . . . . . 9 |- (x = w -> ((ph /\ (x e. X /\ y e. X)) <-> (ph /\ (w e. X /\ y e. X))))
86		opreq2 4890	. . . . . . . . . 10 |- (x = w -> (yHx) = (yHw))
87	42, 86	eqeq12d 1899	. . . . . . . . 9 |- (x = w -> ((xHy) = (yHx) <-> (wHy) = (yHw)))
88	85, 87	imbi12d 688	. . . . . . . 8 |- (x = w -> (((ph /\ (x e. X /\ y e. X)) -> (xHy) = (yHx)) <-> ((ph /\ (w e. X /\ y e. X)) -> (wHy) = (yHw))))
89		iscringd.11	. . . . . . . 8 |- ((ph /\ (x e. X /\ y e. X)) -> (xHy) = (yHx))
90	88, 89	chvarv 1712	. . . . . . 7 |- ((ph /\ (w e. X /\ y e. X)) -> (wHy) = (yHw))
91	83, 90	chvarv 1712	. . . . . 6 |- ((ph /\ (w e. X /\ z e. X)) -> (wHz) = (zHw))
92	69, 76, 91	vtocl 2339	. . . . 5 |- ((ph /\ ((xGy) e. X /\ z e. X)) -> ((xGy)Hz) = (zH(xGy)))
93	68, 92	syldan 516	. . . 4 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy)Hz) = (zH(xGy)))
94	77	anbi2d 678	. . . . . . . . 9 |- (y = z -> ((x e. X /\ y e. X) <-> (x e. X /\ z e. X)))
95	94	anbi2d 678	. . . . . . . 8 |- (y = z -> ((ph /\ (x e. X /\ y e. X)) <-> (ph /\ (x e. X /\ z e. X))))
96		opreq2 4890	. . . . . . . . 9 |- (y = z -> (xHy) = (xHz))
97		opreq1 4889	. . . . . . . . 9 |- (y = z -> (yHx) = (zHx))
98	96, 97	eqeq12d 1899	. . . . . . . 8 |- (y = z -> ((xHy) = (yHx) <-> (xHz) = (zHx)))
99	95, 98	imbi12d 688	. . . . . . 7 |- (y = z -> (((ph /\ (x e. X /\ y e. X)) -> (xHy) = (yHx)) <-> ((ph /\ (x e. X /\ z e. X)) -> (xHz) = (zHx))))
100	99, 89	chvarv 1712	. . . . . 6 |- ((ph /\ (x e. X /\ z e. X)) -> (xHz) = (zHx))
101	100	3adantr2 1036	. . . . 5 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> (xHz) = (zHx))
102		eleq1 1957	. . . . . . . . . . . 12 |- (x = z -> (x e. X <-> z e. X))
103	102	anbi1d 679	. . . . . . . . . . 11 |- (x = z -> ((x e. X /\ y e. X) <-> (z e. X /\ y e. X)))
104	103	anbi2d 678	. . . . . . . . . 10 |- (x = z -> ((ph /\ (x e. X /\ y e. X)) <-> (ph /\ (z e. X /\ y e. X))))
105		opreq1 4889	. . . . . . . . . . 11 |- (x = z -> (xHy) = (zHy))
106		opreq2 4890	. . . . . . . . . . 11 |- (x = z -> (yHx) = (yHz))
107	105, 106	eqeq12d 1899	. . . . . . . . . 10 |- (x = z -> ((xHy) = (yHx) <-> (zHy) = (yHz)))
108	104, 107	imbi12d 688	. . . . . . . . 9 |- (x = z -> (((ph /\ (x e. X /\ y e. X)) -> (xHy) = (yHx)) <-> ((ph /\ (z e. X /\ y e. X)) -> (zHy) = (yHz))))
109	108, 89	chvarv 1712	. . . . . . . 8 |- ((ph /\ (z e. X /\ y e. X)) -> (zHy) = (yHz))
110	109	ancom2s 545	. . . . . . 7 |- ((ph /\ (y e. X /\ z e. X)) -> (zHy) = (yHz))
111	110	eqcomd 1889	. . . . . 6 |- ((ph /\ (y e. X /\ z e. X)) -> (yHz) = (zHy))
112	111	3adantr1 1035	. . . . 5 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> (yHz) = (zHy))
113	101, 112	opreq12d 4900	. . . 4 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xHz)G(yHz)) = ((zHx)G(zHy)))
114	52, 93, 113	3eqtr4d 1937	. . 3 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy)Hz) = ((xHz)G(yHz)))
115		iscringd.9	. . 3 |- (ph -> U e. X)
116		iscringd.10	. . 3 |- ((ph /\ y e. X) -> (yHU) = y)
117	115	adantr 425	. . . . 5 |- ((ph /\ y e. X) -> U e. X)
118		simprl 450	. . . . . . 7 |- ((ph /\ (U e. X /\ y e. X)) -> U e. X)
119		eleq1 1957	. . . . . . . . . . 11 |- (x = U -> (x e. X <-> U e. X))
120	119	anbi1d 679	. . . . . . . . . 10 |- (x = U -> ((x e. X /\ y e. X) <-> (U e. X /\ y e. X)))
121	120	anbi2d 678	. . . . . . . . 9 |- (x = U -> ((ph /\ (x e. X /\ y e. X)) <-> (ph /\ (U e. X /\ y e. X))))
122		opreq1 4889	. . . . . . . . . 10 |- (x = U -> (xHy) = (UHy))
123		opreq2 4890	. . . . . . . . . 10 |- (x = U -> (yHx) = (yHU))
124	122, 123	eqeq12d 1899	. . . . . . . . 9 |- (x = U -> ((xHy) = (yHx) <-> (UHy) = (yHU)))
125	121, 124	imbi12d 688	. . . . . . . 8 |- (x = U -> (((ph /\ (x e. X /\ y e. X)) -> (xHy) = (yHx)) <-> ((ph /\ (U e. X /\ y e. X)) -> (UHy) = (yHU))))
126	125, 89	vtoclg 2346	. . . . . . 7 |- (U e. X -> ((ph /\ (U e. X /\ y e. X)) -> (UHy) = (yHU)))
127	118, 126	mpcom 60	. . . . . 6 |- ((ph /\ (U e. X /\ y e. X)) -> (UHy) = (yHU))
128	127	anass1rs 15646	. . . . 5 |- (((ph /\ y e. X) /\ U e. X) -> (UHy) = (yHU))
129	117, 128	mpdan 768	. . . 4 |- ((ph /\ y e. X) -> (UHy) = (yHU))
130	129, 116	eqtrd 1925	. . 3 |- ((ph /\ y e. X) -> (UHy) = y)
131	1, 2, 5, 6, 7, 8, 9, 10, 114, 115, 116, 130	isringd 16097	. 2 |- (ph -> R e. Ring)
132	89	ex 402	. . . 4 |- (ph -> ((x e. X /\ y e. X) -> (xHy) = (yHx)))
133	55, 132	sylbird 222	. . 3 |- (ph -> ((x e. ran G /\ y e. ran G) -> (xHy) = (yHx)))
134	133	r19.21aivv 2183	. 2 |- (ph -> A.x e. ran GA.y e. ran G(xHy) = (yHx))
135	4, 131, 134	sylanbrc 527	1 |- (ph -> R e. CRing)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   X. cxp 3984  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Grpcgr 9311  Abelcabl 9407  Ringcring 9463  CRingccring 16143

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-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-fo 4012  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-grp 9316  df-abl 9408  df-ring 9464  df-com2 10395  df-cring 16144

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