Theorem isgrp 9321

Description: The predicate "is a group operation." Note that X is the base set of the group.

Hypothesis

Ref	Expression
isgrp.1	|- X = ran G

Assertion

Ref	Expression
isgrp	|- (G e. A -> (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))))

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

Proof of Theorem isgrp

Step	Hyp	Ref	Expression
1		feq1 4551	. . . . . 6 |- (g = G -> (g:(t X. t)-->t <-> G:(t X. t)-->t))
2		opreq 4888	. . . . . . . . . 10 |- (g = G -> ((xgy)gz) = ((xgy)Gz))
3		opreq 4888	. . . . . . . . . . 11 |- (g = G -> (xgy) = (xGy))
4	3	opreq1d 4897	. . . . . . . . . 10 |- (g = G -> ((xgy)Gz) = ((xGy)Gz))
5	2, 4	eqtrd 1925	. . . . . . . . 9 |- (g = G -> ((xgy)gz) = ((xGy)Gz))
6		opreq 4888	. . . . . . . . . 10 |- (g = G -> (xg(ygz)) = (xG(ygz)))
7		opreq 4888	. . . . . . . . . . 11 |- (g = G -> (ygz) = (yGz))
8	7	opreq2d 4898	. . . . . . . . . 10 |- (g = G -> (xG(ygz)) = (xG(yGz)))
9	6, 8	eqtrd 1925	. . . . . . . . 9 |- (g = G -> (xg(ygz)) = (xG(yGz)))
10	5, 9	eqeq12d 1899	. . . . . . . 8 |- (g = G -> (((xgy)gz) = (xg(ygz)) <-> ((xGy)Gz) = (xG(yGz))))
11	10	ralbidv 2123	. . . . . . 7 |- (g = G -> (A.z e. t ((xgy)gz) = (xg(ygz)) <-> A.z e. t ((xGy)Gz) = (xG(yGz))))
12	11	2ralbidv 2140	. . . . . 6 |- (g = G -> (A.x e. t A.y e. t A.z e. t ((xgy)gz) = (xg(ygz)) <-> A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz))))
13		opreq 4888	. . . . . . . . 9 |- (g = G -> (ugx) = (uGx))
14	13	eqeq1d 1892	. . . . . . . 8 |- (g = G -> ((ugx) = x <-> (uGx) = x))
15		opreq 4888	. . . . . . . . . 10 |- (g = G -> (ygx) = (yGx))
16	15	eqeq1d 1892	. . . . . . . . 9 |- (g = G -> ((ygx) = u <-> (yGx) = u))
17	16	rexbidv 2124	. . . . . . . 8 |- (g = G -> (E.y e. t (ygx) = u <-> E.y e. t (yGx) = u))
18	14, 17	anbi12d 690	. . . . . . 7 |- (g = G -> (((ugx) = x /\ E.y e. t (ygx) = u) <-> ((uGx) = x /\ E.y e. t (yGx) = u)))
19	18	rexralbidv 2142	. . . . . 6 |- (g = G -> (E.u e. t A.x e. t ((ugx) = x /\ E.y e. t (ygx) = u) <-> E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u)))
20	1, 12, 19	3anbi123d 1168	. . . . 5 |- (g = G -> ((g:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xgy)gz) = (xg(ygz)) /\ E.u e. t A.x e. t ((ugx) = x /\ E.y e. t (ygx) = u)) <-> (G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u))))
21	20	exbidv 1657	. . . 4 |- (g = G -> (E.t(g:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xgy)gz) = (xg(ygz)) /\ E.u e. t A.x e. t ((ugx) = x /\ E.y e. t (ygx) = u)) <-> E.t(G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u))))
22		df-grp 9316	. . . 4 |- Grp = {g | E.t(g:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xgy)gz) = (xg(ygz)) /\ E.u e. t A.x e. t ((ugx) = x /\ E.y e. t (ygx) = u))}
23	21, 22	elab2g 2406	. . 3 |- (G e. A -> (G e. Grp <-> E.t(G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u))))
24		opreq2 4890	. . . . . . . . . . . . . . . . 17 |- (x = z -> (uGx) = (uGz))
25		id 73	. . . . . . . . . . . . . . . . 17 |- (x = z -> x = z)
26	24, 25	eqeq12d 1899	. . . . . . . . . . . . . . . 16 |- (x = z -> ((uGx) = x <-> (uGz) = z))
27		eqcom 1886	. . . . . . . . . . . . . . . 16 |- ((uGz) = z <-> z = (uGz))
28	26, 27	syl6bb 595	. . . . . . . . . . . . . . 15 |- (x = z -> ((uGx) = x <-> z = (uGz)))
29	28	rcla4v 2376	. . . . . . . . . . . . . 14 |- (z e. t -> (A.x e. t (uGx) = x -> z = (uGz)))
30		opreq2 4890	. . . . . . . . . . . . . . . . 17 |- (y = z -> (uGy) = (uGz))
31	30	eqeq2d 1895	. . . . . . . . . . . . . . . 16 |- (y = z -> (z = (uGy) <-> z = (uGz)))
32	31	rcla4ev 2381	. . . . . . . . . . . . . . 15 |- ((z e. t /\ z = (uGz)) -> E.y e. t z = (uGy))
33	32	ex 402	. . . . . . . . . . . . . 14 |- (z e. t -> (z = (uGz) -> E.y e. t z = (uGy)))
34	29, 33	syld 30	. . . . . . . . . . . . 13 |- (z e. t -> (A.x e. t (uGx) = x -> E.y e. t z = (uGy)))
35		simpl 346	. . . . . . . . . . . . . 14 |- (((uGx) = x /\ E.y e. t (yGx) = u) -> (uGx) = x)
36	35	ralimi 2168	. . . . . . . . . . . . 13 |- (A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u) -> A.x e. t (uGx) = x)
37	34, 36	syl5 20	. . . . . . . . . . . 12 |- (z e. t -> (A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u) -> E.y e. t z = (uGy)))
38	37	reximdv 2202	. . . . . . . . . . 11 |- (z e. t -> (E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u) -> E.u e. t E.y e. t z = (uGy)))
39	38	impcom 378	. . . . . . . . . 10 |- ((E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u) /\ z e. t) -> E.u e. t E.y e. t z = (uGy))
40	39	r19.21aiva 2176	. . . . . . . . 9 |- (E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u) -> A.z e. t E.u e. t E.y e. t z = (uGy))
41	40	anim2i 362	. . . . . . . 8 |- ((G:(t X. t)-->t /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u)) -> (G:(t X. t)-->t /\ A.z e. t E.u e. t E.y e. t z = (uGy)))
42		fooprv 4967	. . . . . . . 8 |- (G:(t X. t)-onto->t <-> (G:(t X. t)-->t /\ A.z e. t E.u e. t E.y e. t z = (uGy)))
43	41, 42	sylibr 217	. . . . . . 7 |- ((G:(t X. t)-->t /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u)) -> G:(t X. t)-onto->t)
44		forn 4620	. . . . . . . 8 |- (G:(t X. t)-onto->t -> ran G = t)
45	44	eqcomd 1889	. . . . . . 7 |- (G:(t X. t)-onto->t -> t = ran G)
46	43, 45	syl 12	. . . . . 6 |- ((G:(t X. t)-->t /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u)) -> t = ran G)
47	46	3adant2 895	. . . . 5 |- ((G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u)) -> t = ran G)
48	47	pm4.71ri 700	. . . 4 |- ((G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u)) <-> (t = ran G /\ (G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u))))
49	48	exbii 1398	. . 3 |- (E.t(G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u)) <-> E.t(t = ran G /\ (G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u))))
50	23, 49	syl6bb 595	. 2 |- (G e. A -> (G e. Grp <-> E.t(t = ran G /\ (G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u)))))
51		rnexg 4207	. . 3 |- (G e. A -> ran G e. _V)
52		isgrp.1	. . . . . 6 |- X = ran G
53	52	eqeq2i 1894	. . . . 5 |- (t = X <-> t = ran G)
54		xpeq1 4016	. . . . . . . . 9 |- (t = X -> (t X. t) = (X X. t))
55		xpeq2 4017	. . . . . . . . 9 |- (t = X -> (X X. t) = (X X. X))
56	54, 55	eqtrd 1925	. . . . . . . 8 |- (t = X -> (t X. t) = (X X. X))
57	56	feq2d 4557	. . . . . . 7 |- (t = X -> (G:(t X. t)-->t <-> G:(X X. X)-->t))
58		feq3 4553	. . . . . . 7 |- (t = X -> (G:(X X. X)-->t <-> G:(X X. X)-->X))
59	57, 58	bitrd 587	. . . . . 6 |- (t = X -> (G:(t X. t)-->t <-> G:(X X. X)-->X))
60		raleq 2266	. . . . . . . 8 |- (t = X -> (A.z e. t ((xGy)Gz) = (xG(yGz)) <-> A.z e. X ((xGy)Gz) = (xG(yGz))))
61	60	raleqbi1dv 2271	. . . . . . 7 |- (t = X -> (A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) <-> A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))))
62	61	raleqbi1dv 2271	. . . . . 6 |- (t = X -> (A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) <-> A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))))
63		rexeq 2267	. . . . . . . . 9 |- (t = X -> (E.y e. t (yGx) = u <-> E.y e. X (yGx) = u))
64	63	anbi2d 678	. . . . . . . 8 |- (t = X -> (((uGx) = x /\ E.y e. t (yGx) = u) <-> ((uGx) = x /\ E.y e. X (yGx) = u)))
65	64	raleqbi1dv 2271	. . . . . . 7 |- (t = X -> (A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u) <-> A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)))
66	65	rexeqbi1dv 2272	. . . . . 6 |- (t = X -> (E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u) <-> E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)))
67	59, 62, 66	3anbi123d 1168	. . . . 5 |- (t = X -> ((G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u)) <-> (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))))
68	53, 67	sylbir 218	. . . 4 |- (t = ran G -> ((G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u)) <-> (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))))
69	68	ceqsexgv 2393	. . 3 |- (ran G e. _V -> (E.t(t = ran G /\ (G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u))) <-> (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))))
70	51, 69	syl 12	. 2 |- (G e. A -> (E.t(t = ran G /\ (G:(t X. t)-->t /\ A.x e. t A.y e. t A.z e. t ((xGy)Gz) = (xG(yGz)) /\ E.u e. t A.x e. t ((uGx) = x /\ E.y e. t (yGx) = u))) <-> (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))))
71	50, 70	bitrd 587	1 |- (G e. A -> (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))))

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

This theorem is referenced by:  isgrpi 9322  grpfo 9323  grplidinv 9325  grpass 9327  isgrp2i 9360  grpmnd 10393  hmeogrp 14892  isgrpda 16033

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

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