Theorem isgrpi 9322

Description: Properties that determine a group operation. Read N as N(x).

Hypotheses

Ref	Expression
isgrpi.1	|- X e. _V
isgrpi.2	|- G:(X X. X)-->X
isgrpi.3	|- ((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz)))
isgrpi.4	|- U e. X
isgrpi.5	|- (x e. X -> (UGx) = x)
isgrpi.6	|- (x e. X -> N e. X)
isgrpi.7	|- (x e. X -> (NGx) = U)

Assertion

Ref	Expression
isgrpi	|- G e. Grp

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

Proof of Theorem isgrpi

Step	Hyp	Ref	Expression
1		isgrpi.2	. . . 4 |- G:(X X. X)-->X
2		isgrpi.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.x e. X E.y e. X E.z e. X x = (yGz)))
7		isgrpi.5	. . . . . . . . . 10 |- (x e. X -> (UGx) = x)
8	7	eqcomd 1889	. . . . . . . . 9 |- (x e. X -> x = (UGx))
9		isgrpi.4	. . . . . . . . . 10 |- U e. X
10		rcla4eopr 4915	. . . . . . . . . 10 |- ((U e. X /\ x e. X /\ x = (UGx)) -> E.y e. X E.z e. X x = (yGz))
11	9, 10	mp3an1 1178	. . . . . . . . 9 |- ((x e. X /\ x = (UGx)) -> E.y e. X E.z e. X x = (yGz))
12	8, 11	mpdan 768	. . . . . . . 8 |- (x e. X -> E.y e. X E.z e. X x = (yGz))
13	12	rgen 2159	. . . . . . 7 |- A.x e. X E.y e. X E.z e. X x = (yGz)
14	6, 1, 13	mpbir2an 800	. . . . . 6 |- G:(X X. X)-onto->X
15		forn 4620	. . . . . 6 |- (G:(X X. X)-onto->X -> ran G = X)
16	14, 15	ax-mp 7	. . . . 5 |- ran G = X
17	16	eqcomi 1888	. . . 4 |- X = ran G
18	17	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))))
19	5, 18	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)))
20		isgrpi.3	. . 3 |- ((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz)))
21	20	rgen3 2187	. 2 |- A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))
22		isgrpi.6	. . . . . 6 |- (x e. X -> N e. X)
23		isgrpi.7	. . . . . 6 |- (x e. X -> (NGx) = U)
24		opreq1 4889	. . . . . . . 8 |- (y = N -> (yGx) = (NGx))
25	24	eqeq1d 1892	. . . . . . 7 |- (y = N -> ((yGx) = U <-> (NGx) = U))
26	25	rcla4ev 2381	. . . . . 6 |- ((N e. X /\ (NGx) = U) -> E.y e. X (yGx) = U)
27	22, 23, 26	syl11anc 524	. . . . 5 |- (x e. X -> E.y e. X (yGx) = U)
28	7, 27	jca 310	. . . 4 |- (x e. X -> ((UGx) = x /\ E.y e. X (yGx) = U))
29	28	rgen 2159	. . 3 |- A.x e. X ((UGx) = x /\ E.y e. X (yGx) = U)
30		opreq1 4889	. . . . . . 7 |- (u = U -> (uGx) = (UGx))
31	30	eqeq1d 1892	. . . . . 6 |- (u = U -> ((uGx) = x <-> (UGx) = x))
32		eqeq2 1893	. . . . . . 7 |- (u = U -> ((yGx) = u <-> (yGx) = U))
33	32	rexbidv 2124	. . . . . 6 |- (u = U -> (E.y e. X (yGx) = u <-> E.y e. X (yGx) = U))
34	31, 33	anbi12d 690	. . . . 5 |- (u = U -> (((uGx) = x /\ E.y e. X (yGx) = u) <-> ((UGx) = x /\ E.y e. X (yGx) = U)))
35	34	ralbidv 2123	. . . 4 |- (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)))
36	35	rcla4ev 2381	. . 3 |- ((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))
37	9, 29, 36	mp2an 761	. 2 |- E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)
38	19, 1, 21, 37	mpbir3an 1052	1 |- 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:  grpsn 9340  issubgi 9431  cnaddabl 9434  ablmul 9439  symggrpi 10205  hilabl 10660

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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