Theorem isgrp2i 9360

Description: An alternate way to show a group operation. Exercise 1 of [Herstein] p. 57.

Hypotheses

Ref	Expression
isgrp2i.1	|- X e. _V
isgrp2i.2	|- X =/= (/)
isgrp2i.3	|- G:(X X. X)-->X
isgrp2i.4	|- ((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz)))
isgrp2i.5	|- ((x e. X /\ y e. X) -> E.z e. X (zGx) = y)
isgrp2i.6	|- ((x e. X /\ y e. X) -> E.z e. X (xGz) = y)

Assertion

Ref	Expression
isgrp2i	|- G e. Grp

Distinct variable groups:   x,y,z,G   x,X,y,z

Proof of Theorem isgrp2i

Step	Hyp	Ref	Expression
1		isgrp2i.3	. . . 4 |- G:(X X. X)-->X
2		isgrp2i.1	. . . . 5 |- X e. _V
3	2, 2	xpex 4096	. . . 4 |- (X X. X) e. _V
4		fex 4595	. . . 4 |- ((G:(X X. X)-->X /\ (X X. X) e. _V) -> G e. _V)
5	1, 3, 4	mp2an 761	. . 3 |- G e. _V
6		fooprv 4967	. . . . . . 7 |- (G:(X X. X)-onto->X <-> (G:(X X. X)-->X /\ A.y e. X E.x e. X E.z e. X y = (xGz)))
7		r19.2z 2958	. . . . . . . . 9 |- ((X =/= (/) /\ A.x e. X E.z e. X y = (xGz)) -> E.x e. X E.z e. X y = (xGz))
8		isgrp2i.2	. . . . . . . . 9 |- X =/= (/)
9		isgrp2i.6	. . . . . . . . . . . 12 |- ((x e. X /\ y e. X) -> E.z e. X (xGz) = y)
10	9	ancoms 484	. . . . . . . . . . 11 |- ((y e. X /\ x e. X) -> E.z e. X (xGz) = y)
11		eqcom 1886	. . . . . . . . . . . 12 |- ((xGz) = y <-> y = (xGz))
12	11	rexbii 2128	. . . . . . . . . . 11 |- (E.z e. X (xGz) = y <-> E.z e. X y = (xGz))
13	10, 12	sylib 215	. . . . . . . . . 10 |- ((y e. X /\ x e. X) -> E.z e. X y = (xGz))
14	13	r19.21aiva 2176	. . . . . . . . 9 |- (y e. X -> A.x e. X E.z e. X y = (xGz))
15	7, 8, 14	sylancr 526	. . . . . . . 8 |- (y e. X -> E.x e. X E.z e. X y = (xGz))
16	15	rgen 2159	. . . . . . 7 |- A.y e. X E.x e. X E.z e. X y = (xGz)
17	6, 1, 16	mpbir2an 800	. . . . . 6 |- G:(X X. X)-onto->X
18		forn 4620	. . . . . 6 |- (G:(X X. X)-onto->X -> ran G = X)
19	17, 18	ax-mp 7	. . . . 5 |- ran G = X
20	19	eqcomi 1888	. . . 4 |- X = ran G
21	20	isgrp 9321	. . 3 |- (G e. _V -> (G e. Grp <-> (G:(X X. X)-->X /\ A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz)) /\ E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u))))
22	5, 21	ax-mp 7	. 2 |- (G e. Grp <-> (G:(X X. X)-->X /\ A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz)) /\ E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)))
23		isgrp2i.4	. . 3 |- ((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz)))
24	23	rgen3 2187	. 2 |- A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))
25		n0 2884	. . . 4 |- (X =/= (/) <-> E.w w e. X)
26	8, 25	mpbi 206	. . 3 |- E.w w e. X
27		eleq1 1957	. . . . . . . . . 10 |- (x = w -> (x e. X <-> w e. X))
28	27	anbi1d 679	. . . . . . . . 9 |- (x = w -> ((x e. X /\ w e. X) <-> (w e. X /\ w e. X)))
29		opreq2 4890	. . . . . . . . . . 11 |- (x = w -> (zGx) = (zGw))
30	29	eqeq1d 1892	. . . . . . . . . 10 |- (x = w -> ((zGx) = w <-> (zGw) = w))
31	30	rexbidv 2124	. . . . . . . . 9 |- (x = w -> (E.z e. X (zGx) = w <-> E.z e. X (zGw) = w))
32	28, 31	imbi12d 688	. . . . . . . 8 |- (x = w -> (((x e. X /\ w e. X) -> E.z e. X (zGx) = w) <-> ((w e. X /\ w e. X) -> E.z e. X (zGw) = w)))
33		eleq1 1957	. . . . . . . . . . 11 |- (y = w -> (y e. X <-> w e. X))
34	33	anbi2d 678	. . . . . . . . . 10 |- (y = w -> ((x e. X /\ y e. X) <-> (x e. X /\ w e. X)))
35		eqeq2 1893	. . . . . . . . . . 11 |- (y = w -> ((zGx) = y <-> (zGx) = w))
36	35	rexbidv 2124	. . . . . . . . . 10 |- (y = w -> (E.z e. X (zGx) = y <-> E.z e. X (zGx) = w))
37	34, 36	imbi12d 688	. . . . . . . . 9 |- (y = w -> (((x e. X /\ y e. X) -> E.z e. X (zGx) = y) <-> ((x e. X /\ w e. X) -> E.z e. X (zGx) = w)))
38		isgrp2i.5	. . . . . . . . 9 |- ((x e. X /\ y e. X) -> E.z e. X (zGx) = y)
39	37, 38	chvarv 1712	. . . . . . . 8 |- ((x e. X /\ w e. X) -> E.z e. X (zGx) = w)
40	32, 39	chvarv 1712	. . . . . . 7 |- ((w e. X /\ w e. X) -> E.z e. X (zGw) = w)
41	40	anidms 480	. . . . . 6 |- (w e. X -> E.z e. X (zGw) = w)
42		opreq1 4889	. . . . . . . 8 |- (z = u -> (zGw) = (uGw))
43	42	eqeq1d 1892	. . . . . . 7 |- (z = u -> ((zGw) = w <-> (uGw) = w))
44	43	cbvrexv 2281	. . . . . 6 |- (E.z e. X (zGw) = w <-> E.u e. X (uGw) = w)
45	41, 44	sylib 215	. . . . 5 |- (w e. X -> E.u e. X (uGw) = w)
46		opreq1 4889	. . . . . . . . . . . . . . . . 17 |- (x = w -> (xGz) = (wGz))
47	46	eqeq1d 1892	. . . . . . . . . . . . . . . 16 |- (x = w -> ((xGz) = y <-> (wGz) = y))
48	47	rexbidv 2124	. . . . . . . . . . . . . . 15 |- (x = w -> (E.z e. X (xGz) = y <-> E.z e. X (wGz) = y))
49		eqeq2 1893	. . . . . . . . . . . . . . . 16 |- (y = x -> ((wGz) = y <-> (wGz) = x))
50	49	rexbidv 2124	. . . . . . . . . . . . . . 15 |- (y = x -> (E.z e. X (wGz) = y <-> E.z e. X (wGz) = x))
51	48, 50, 9	vtocl2ga 2353	. . . . . . . . . . . . . 14 |- ((w e. X /\ x e. X) -> E.z e. X (wGz) = x)
52		opreq2 4890	. . . . . . . . . . . . . . . 16 |- (z = v -> (wGz) = (wGv))
53	52	eqeq1d 1892	. . . . . . . . . . . . . . 15 |- (z = v -> ((wGz) = x <-> (wGv) = x))
54	53	cbvrexv 2281	. . . . . . . . . . . . . 14 |- (E.z e. X (wGz) = x <-> E.v e. X (wGv) = x)
55	51, 54	sylib 215	. . . . . . . . . . . . 13 |- ((w e. X /\ x e. X) -> E.v e. X (wGv) = x)
56	55	adantlr 429	. . . . . . . . . . . 12 |- (((w e. X /\ u e. X) /\ x e. X) -> E.v e. X (wGv) = x)
57	56	adantr 425	. . . . . . . . . . 11 |- ((((w e. X /\ u e. X) /\ x e. X) /\ (uGw) = w) -> E.v e. X (wGv) = x)
58		eleq1 1957	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x = u -> (x e. X <-> u e. X))
59	58	3anbi1d 1172	. . . . . . . . . . . . . . . . . . . . . 22 |- (x = u -> ((x e. X /\ w e. X /\ v e. X) <-> (u e. X /\ w e. X /\ v e. X)))
60		opreq1 4889	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (x = u -> (xGw) = (uGw))
61	60	opreq1d 4897	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x = u -> ((xGw)Gv) = ((uGw)Gv))
62		opreq1 4889	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x = u -> (xG(wGv)) = (uG(wGv)))
63	61, 62	eqeq12d 1899	. . . . . . . . . . . . . . . . . . . . . 22 |- (x = u -> (((xGw)Gv) = (xG(wGv)) <-> ((uGw)Gv) = (uG(wGv))))
64	59, 63	imbi12d 688	. . . . . . . . . . . . . . . . . . . . 21 |- (x = u -> (((x e. X /\ w e. X /\ v e. X) -> ((xGw)Gv) = (xG(wGv))) <-> ((u e. X /\ w e. X /\ v e. X) -> ((uGw)Gv) = (uG(wGv)))))
65	33	3anbi2d 1173	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y = w -> ((x e. X /\ y e. X /\ v e. X) <-> (x e. X /\ w e. X /\ v e. X)))
66		opreq2 4890	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y = w -> (xGy) = (xGw))
67	66	opreq1d 4897	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y = w -> ((xGy)Gv) = ((xGw)Gv))
68		opreq1 4889	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y = w -> (yGv) = (wGv))
69	68	opreq2d 4898	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y = w -> (xG(yGv)) = (xG(wGv)))
70	67, 69	eqeq12d 1899	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y = w -> (((xGy)Gv) = (xG(yGv)) <-> ((xGw)Gv) = (xG(wGv))))
71	65, 70	imbi12d 688	. . . . . . . . . . . . . . . . . . . . . 22 |- (y = w -> (((x e. X /\ y e. X /\ v e. X) -> ((xGy)Gv) = (xG(yGv))) <-> ((x e. X /\ w e. X /\ v e. X) -> ((xGw)Gv) = (xG(wGv)))))
72		eleq1 1957	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (z = v -> (z e. X <-> v e. X))
73	72	3anbi3d 1174	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (z = v -> ((x e. X /\ y e. X /\ z e. X) <-> (x e. X /\ y e. X /\ v e. X)))
74		opreq2 4890	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (z = v -> ((xGy)Gz) = ((xGy)Gv))
75		opreq2 4890	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (z = v -> (yGz) = (yGv))
76	75	opreq2d 4898	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (z = v -> (xG(yGz)) = (xG(yGv)))
77	74, 76	eqeq12d 1899	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (z = v -> (((xGy)Gz) = (xG(yGz)) <-> ((xGy)Gv) = (xG(yGv))))
78	73, 77	imbi12d 688	. . . . . . . . . . . . . . . . . . . . . . 23 |- (z = v -> (((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz))) <-> ((x e. X /\ y e. X /\ v e. X) -> ((xGy)Gv) = (xG(yGv)))))
79	78, 23	chvarv 1712	. . . . . . . . . . . . . . . . . . . . . 22 |- ((x e. X /\ y e. X /\ v e. X) -> ((xGy)Gv) = (xG(yGv)))
80	71, 79	chvarv 1712	. . . . . . . . . . . . . . . . . . . . 21 |- ((x e. X /\ w e. X /\ v e. X) -> ((xGw)Gv) = (xG(wGv)))
81	64, 80	chvarv 1712	. . . . . . . . . . . . . . . . . . . 20 |- ((u e. X /\ w e. X /\ v e. X) -> ((uGw)Gv) = (uG(wGv)))
82	81	3com12 1071	. . . . . . . . . . . . . . . . . . 19 |- ((w e. X /\ u e. X /\ v e. X) -> ((uGw)Gv) = (uG(wGv)))
83	82	3expa 1067	. . . . . . . . . . . . . . . . . 18 |- (((w e. X /\ u e. X) /\ v e. X) -> ((uGw)Gv) = (uG(wGv)))
84	83	adantlr 429	. . . . . . . . . . . . . . . . 17 |- ((((w e. X /\ u e. X) /\ (uGw) = w) /\ v e. X) -> ((uGw)Gv) = (uG(wGv)))
85		opreq1 4889	. . . . . . . . . . . . . . . . . 18 |- ((uGw) = w -> ((uGw)Gv) = (wGv))
86	85	ad2antlr 441	. . . . . . . . . . . . . . . . 17 |- ((((w e. X /\ u e. X) /\ (uGw) = w) /\ v e. X) -> ((uGw)Gv) = (wGv))
87	84, 86	eqtr3d 1927	. . . . . . . . . . . . . . . 16 |- ((((w e. X /\ u e. X) /\ (uGw) = w) /\ v e. X) -> (uG(wGv)) = (wGv))
88	87	adantr 425	. . . . . . . . . . . . . . 15 |- (((((w e. X /\ u e. X) /\ (uGw) = w) /\ v e. X) /\ (wGv) = x) -> (uG(wGv)) = (wGv))
89		opreq2 4890	. . . . . . . . . . . . . . . 16 |- ((wGv) = x -> (uG(wGv)) = (uGx))
90	89	adantl 424	. . . . . . . . . . . . . . 15 |- (((((w e. X /\ u e. X) /\ (uGw) = w) /\ v e. X) /\ (wGv) = x) -> (uG(wGv)) = (uGx))
91		simpr 350	. . . . . . . . . . . . . . 15 |- (((((w e. X /\ u e. X) /\ (uGw) = w) /\ v e. X) /\ (wGv) = x) -> (wGv) = x)
92	88, 90, 91	3eqtr3d 1934	. . . . . . . . . . . . . 14 |- (((((w e. X /\ u e. X) /\ (uGw) = w) /\ v e. X) /\ (wGv) = x) -> (uGx) = x)
93	92	exp31 407	. . . . . . . . . . . . 13 |- (((w e. X /\ u e. X) /\ (uGw) = w) -> (v e. X -> ((wGv) = x -> (uGx) = x)))
94	93	r19.23adv 2215	. . . . . . . . . . . 12 |- (((w e. X /\ u e. X) /\ (uGw) = w) -> (E.v e. X (wGv) = x -> (uGx) = x))
95	94	adantlr 429	. . . . . . . . . . 11 |- ((((w e. X /\ u e. X) /\ x e. X) /\ (uGw) = w) -> (E.v e. X (wGv) = x -> (uGx) = x))
96	57, 95	mpd 29	. . . . . . . . . 10 |- ((((w e. X /\ u e. X) /\ x e. X) /\ (uGw) = w) -> (uGx) = x)
97		eleq1 1957	. . . . . . . . . . . . . . . . 17 |- (y = u -> (y e. X <-> u e. X))
98	97	anbi2d 678	. . . . . . . . . . . . . . . 16 |- (y = u -> ((x e. X /\ y e. X) <-> (x e. X /\ u e. X)))
99		eqeq2 1893	. . . . . . . . . . . . . . . . 17 |- (y = u -> ((zGx) = y <-> (zGx) = u))
100	99	rexbidv 2124	. . . . . . . . . . . . . . . 16 |- (y = u -> (E.z e. X (zGx) = y <-> E.z e. X (zGx) = u))
101	98, 100	imbi12d 688	. . . . . . . . . . . . . . 15 |- (y = u -> (((x e. X /\ y e. X) -> E.z e. X (zGx) = y) <-> ((x e. X /\ u e. X) -> E.z e. X (zGx) = u)))
102	101, 38	chvarv 1712	. . . . . . . . . . . . . 14 |- ((x e. X /\ u e. X) -> E.z e. X (zGx) = u)
103		opreq1 4889	. . . . . . . . . . . . . . . 16 |- (z = y -> (zGx) = (yGx))
104	103	eqeq1d 1892	. . . . . . . . . . . . . . 15 |- (z = y -> ((zGx) = u <-> (yGx) = u))
105	104	cbvrexv 2281	. . . . . . . . . . . . . 14 |- (E.z e. X (zGx) = u <-> E.y e. X (yGx) = u)
106	102, 105	sylib 215	. . . . . . . . . . . . 13 |- ((x e. X /\ u e. X) -> E.y e. X (yGx) = u)
107	106	ancoms 484	. . . . . . . . . . . 12 |- ((u e. X /\ x e. X) -> E.y e. X (yGx) = u)
108	107	adantll 428	. . . . . . . . . . 11 |- (((w e. X /\ u e. X) /\ x e. X) -> E.y e. X (yGx) = u)
109	108	adantr 425	. . . . . . . . . 10 |- ((((w e. X /\ u e. X) /\ x e. X) /\ (uGw) = w) -> E.y e. X (yGx) = u)
110	96, 109	jca 310	. . . . . . . . 9 |- ((((w e. X /\ u e. X) /\ x e. X) /\ (uGw) = w) -> ((uGx) = x /\ E.y e. X (yGx) = u))
111	110	exp31 407	. . . . . . . 8 |- ((w e. X /\ u e. X) -> (x e. X -> ((uGw) = w -> ((uGx) = x /\ E.y e. X (yGx) = u))))
112	111	com23 36	. . . . . . 7 |- ((w e. X /\ u e. X) -> ((uGw) = w -> (x e. X -> ((uGx) = x /\ E.y e. X (yGx) = u))))
113	112	r19.21adv 2181	. . . . . 6 |- ((w e. X /\ u e. X) -> ((uGw) = w -> A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)))
114	113	reximdva 2203	. . . . 5 |- (w e. X -> (E.u e. X (uGw) = w -> E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)))
115	45, 114	mpd 29	. . . 4 |- (w e. X -> E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u))
116	115	19.23aiv 1674	. . 3 |- (E.w w e. X -> E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u))
117	26, 116	ax-mp 7	. 2 |- E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)
118	22, 1, 24, 117	mpbir3an 1052	1 |- G e. Grp

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292  (/)c0 2875   X. cxp 3984  ran crn 3987  -->wf 3994  -onto->wfo 3996  (class class class)co 4884  Grpcgr 9311

This theorem is referenced by:  ghgrpilem4 9444

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-rep 3428  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-fo 4012  df-fv 4014  df-opr 4886  df-grp 9316

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