Theorem isgrpda 16033

Description: Properties that determine a group operation.

Hypotheses

Ref	Expression
isgrpda.1	|- (ph -> X e. _V)
isgrpda.2	|- (ph -> G:(X X. X)-->X)
isgrpda.3	|- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy)Gz) = (xG(yGz)))
isgrpda.4	|- (ph -> U e. X)
isgrpda.5	|- ((ph /\ x e. X) -> (UGx) = x)
isgrpda.6	|- ((ph /\ x e. X) -> E.n e. X (nGx) = U)

Assertion

Ref	Expression
isgrpda	|- (ph -> G e. Grp)

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

Proof of Theorem isgrpda

Step	Hyp	Ref	Expression
1		isgrpda.2	. . . 4 |- (ph -> G:(X X. X)-->X)
2		isgrpda.3	. . . . . . . 8 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy)Gz) = (xG(yGz)))
3	2	3exp2 1086	. . . . . . 7 |- (ph -> (x e. X -> (y e. X -> (z e. X -> ((xGy)Gz) = (xG(yGz))))))
4	3	imp3a 388	. . . . . 6 |- (ph -> ((x e. X /\ y e. X) -> (z e. X -> ((xGy)Gz) = (xG(yGz)))))
5	4	r19.21adv 2181	. . . . 5 |- (ph -> ((x e. X /\ y e. X) -> A.z e. X ((xGy)Gz) = (xG(yGz))))
6	5	r19.21aivv 2183	. . . 4 |- (ph -> A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz)))
7		isgrpda.4	. . . . 5 |- (ph -> U e. X)
8		isgrpda.5	. . . . . . 7 |- ((ph /\ x e. X) -> (UGx) = x)
9		isgrpda.6	. . . . . . . 8 |- ((ph /\ x e. X) -> E.n e. X (nGx) = U)
10		opreq1 4889	. . . . . . . . . 10 |- (y = n -> (yGx) = (nGx))
11	10	eqeq1d 1892	. . . . . . . . 9 |- (y = n -> ((yGx) = U <-> (nGx) = U))
12	11	cbvrexv 2281	. . . . . . . 8 |- (E.y e. X (yGx) = U <-> E.n e. X (nGx) = U)
13	9, 12	sylibr 217	. . . . . . 7 |- ((ph /\ x e. X) -> E.y e. X (yGx) = U)
14	8, 13	jca 310	. . . . . 6 |- ((ph /\ x e. X) -> ((UGx) = x /\ E.y e. X (yGx) = U))
15	14	r19.21aiva 2176	. . . . 5 |- (ph -> A.x e. X ((UGx) = x /\ E.y e. X (yGx) = U))
16		opreq1 4889	. . . . . . . . 9 |- (u = U -> (uGx) = (UGx))
17	16	eqeq1d 1892	. . . . . . . 8 |- (u = U -> ((uGx) = x <-> (UGx) = x))
18		eqeq2 1893	. . . . . . . . 9 |- (u = U -> ((yGx) = u <-> (yGx) = U))
19	18	rexbidv 2124	. . . . . . . 8 |- (u = U -> (E.y e. X (yGx) = u <-> E.y e. X (yGx) = U))
20	17, 19	anbi12d 690	. . . . . . 7 |- (u = U -> (((uGx) = x /\ E.y e. X (yGx) = u) <-> ((UGx) = x /\ E.y e. X (yGx) = U)))
21	20	ralbidv 2123	. . . . . 6 |- (u = U -> (A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u) <-> A.x e. X ((UGx) = x /\ E.y e. X (yGx) = U)))
22	21	rcla4ev 2381	. . . . 5 |- ((U e. X /\ A.x e. X ((UGx) = x /\ E.y e. X (yGx) = U)) -> E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u))
23	7, 15, 22	syl11anc 524	. . . 4 |- (ph -> E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u))
24	1, 6, 23	3jca 1050	. . 3 |- (ph -> (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)))
25		fooprv 4967	. . . . . . . . 9 |- (G:(X X. X)-onto->X <-> (G:(X X. X)-->X /\ A.x e. X E.y e. X E.z e. X x = (yGz)))
26	7	adantr 425	. . . . . . . . . . 11 |- ((ph /\ x e. X) -> U e. X)
27		simpr 350	. . . . . . . . . . 11 |- ((ph /\ x e. X) -> x e. X)
28	8	eqcomd 1889	. . . . . . . . . . 11 |- ((ph /\ x e. X) -> x = (UGx))
29		rcla4eopr 4915	. . . . . . . . . . 11 |- ((U e. X /\ x e. X /\ x = (UGx)) -> E.y e. X E.z e. X x = (yGz))
30	26, 27, 28, 29	syl111anc 1100	. . . . . . . . . 10 |- ((ph /\ x e. X) -> E.y e. X E.z e. X x = (yGz))
31	30	r19.21aiva 2176	. . . . . . . . 9 |- (ph -> A.x e. X E.y e. X E.z e. X x = (yGz))
32	25, 1, 31	sylanbrc 527	. . . . . . . 8 |- (ph -> G:(X X. X)-onto->X)
33		forn 4620	. . . . . . . 8 |- (G:(X X. X)-onto->X -> ran G = X)
34		xpeq1 4016	. . . . . . . 8 |- (ran G = X -> (ran G X. ran G) = (X X. ran G))
35	32, 33, 34	3syl 24	. . . . . . 7 |- (ph -> (ran G X. ran G) = (X X. ran G))
36		xpeq2 4017	. . . . . . . 8 |- (ran G = X -> (X X. ran G) = (X X. X))
37	32, 33, 36	3syl 24	. . . . . . 7 |- (ph -> (X X. ran G) = (X X. X))
38	35, 37	eqtrd 1925	. . . . . 6 |- (ph -> (ran G X. ran G) = (X X. X))
39	38	feq2d 4557	. . . . 5 |- (ph -> (G:(ran G X. ran G)-->ran G <-> G:(X X. X)-->ran G))
40		feq3 4553	. . . . . 6 |- (ran G = X -> (G:(X X. X)-->ran G <-> G:(X X. X)-->X))
41	32, 33, 40	3syl 24	. . . . 5 |- (ph -> (G:(X X. X)-->ran G <-> G:(X X. X)-->X))
42	39, 41	bitrd 587	. . . 4 |- (ph -> (G:(ran G X. ran G)-->ran G <-> G:(X X. X)-->X))
43	32, 33	syl 12	. . . . 5 |- (ph -> ran G = X)
44	43	raleqdv 2269	. . . . . 6 |- (ph -> (A.z e. ran G((xGy)Gz) = (xG(yGz)) <-> A.z e. X ((xGy)Gz) = (xG(yGz))))
45	43, 44	raleqbidv 2274	. . . . 5 |- (ph -> (A.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) <-> A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))))
46	43, 45	raleqbidv 2274	. . . 4 |- (ph -> (A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) <-> A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))))
47	43	rexeqdv 2270	. . . . . . 7 |- (ph -> (E.y e. ran G(yGx) = u <-> E.y e. X (yGx) = u))
48	47	anbi2d 678	. . . . . 6 |- (ph -> (((uGx) = x /\ E.y e. ran G(yGx) = u) <-> ((uGx) = x /\ E.y e. X (yGx) = u)))
49	43, 48	raleqbidv 2274	. . . . 5 |- (ph -> (A.x e. ran G((uGx) = x /\ E.y e. ran G(yGx) = u) <-> A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)))
50	43, 49	rexeqbidv 2275	. . . 4 |- (ph -> (E.u e. ran GA.x e. ran G((uGx) = x /\ E.y e. ran G(yGx) = u) <-> E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)))
51	42, 46, 50	3anbi123d 1168	. . 3 |- (ph -> ((G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.u e. ran GA.x e. ran G((uGx) = x /\ E.y e. ran G(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))))
52	24, 51	mpbird 213	. 2 |- (ph -> (G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.u e. ran GA.x e. ran G((uGx) = x /\ E.y e. ran G(yGx) = u)))
53		isgrpda.1	. . . . 5 |- (ph -> X e. _V)
54		xpexg 4095	. . . . 5 |- ((X e. _V /\ X e. _V) -> (X X. X) e. _V)
55	53, 53, 54	syl11anc 524	. . . 4 |- (ph -> (X X. X) e. _V)
56		fex 4595	. . . 4 |- ((G:(X X. X)-->X /\ (X X. X) e. _V) -> G e. _V)
57	1, 55, 56	syl11anc 524	. . 3 |- (ph -> G e. _V)
58		eqid 1884	. . . 4 |- ran G = ran G
59	58	isgrp 9321	. . 3 |- (G e. _V -> (G e. Grp <-> (G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.u e. ran GA.x e. ran G((uGx) = x /\ E.y e. ran G(yGx) = u))))
60	57, 59	syl 12	. 2 |- (ph -> (G e. Grp <-> (G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.u e. ran GA.x e. ran G((uGx) = x /\ E.y e. ran G(yGx) = u))))
61	52, 60	mpbird 213	1 |- (ph -> G e. Grp)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  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:  isgrpd 16034  isablda 16035  pi1gp 16095  isdivrng2 16111

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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