Theorem isgrpi 8042

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	. . 3 |- G:(X X. X)-->X
2		isgrpi.3	. . . 4 |- ((x e. X /\ y e. X /\ z e. X) -> ((xGy)Gz) = (xG(yGz)))
3	2	rgen3 1724	. . 3 |- A.x e. X A.y e. X A.z e. X ((xGy)Gz) = (xG(yGz))
4		isgrpi.4	. . . 4 |- U e. X
5		isgrpi.5	. . . . . 6 |- (x e. X -> (UGx) = x)
6		opreq1 3968	. . . . . . . . 9 |- (y = N -> (yGx) = (NGx))
7	6	eqeq1d 1483	. . . . . . . 8 |- (y = N -> ((yGx) = U <-> (NGx) = U))
8	7	rcla4ev 1877	. . . . . . 7 |- ((N e. X /\ (NGx) = U) -> E.y e. X (yGx) = U)
9		isgrpi.6	. . . . . . 7 |- (x e. X -> N e. X)
10		isgrpi.7	. . . . . . 7 |- (x e. X -> (NGx) = U)
11	8, 9, 10	sylanc 471	. . . . . 6 |- (x e. X -> E.y e. X (yGx) = U)
12	5, 11	jca 288	. . . . 5 |- (x e. X -> ((UGx) = x /\ E.y e. X (yGx) = U))
13	12	rgen 1698	. . . 4 |- A.x e. X ((UGx) = x /\ E.y e. X (yGx) = U)
14		opreq1 3968	. . . . . . . 8 |- (u = U -> (uGx) = (UGx))
15	14	eqeq1d 1483	. . . . . . 7 |- (u = U -> ((uGx) = x <-> (UGx) = x))
16		eqeq2 1484	. . . . . . . 8 |- (u = U -> ((yGx) = u <-> (yGx) = U))
17	16	rexbidv 1664	. . . . . . 7 |- (u = U -> (E.y e. X (yGx) = u <-> E.y e. X (yGx) = U))
18	15, 17	anbi12d 628	. . . . . 6 |- (u = U -> (((uGx) = x /\ E.y e. X (yGx) = u) <-> ((UGx) = x /\ E.y e. X (yGx) = U)))
19	18	ralbidv 1663	. . . . 5 |- (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)))
20	19	rcla4ev 1877	. . . 4 |- ((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))
21	4, 13, 20	mp2an 697	. . 3 |- E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)
22	1, 3, 21	3pm3.2i 818	. 2 |- (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		isgrpi.1	. . . . 5 |- X e. V
24	23, 23	xpex 3260	. . . 4 |- (X X. X) e. V
25		fex 3652	. . . 4 |- ((G:(X X. X)-->X /\ (X X. X) e. V) -> G e. V)
26	1, 24, 25	mp2an 697	. . 3 |- G e. V
27		fooprval 4037	. . . . . . 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)))
28	5	eqcomd 1480	. . . . . . . . 9 |- (x e. X -> x = (UGx))
29		rcla4eopr 3990	. . . . . . . . . 10 |- ((U e. X /\ x e. X /\ x = (UGx)) -> E.y e. X E.z e. X x = (yGz))
30	4, 29	mp3an1 903	. . . . . . . . 9 |- ((x e. X /\ x = (UGx)) -> E.y e. X E.z e. X x = (yGz))
31	28, 30	mpdan 704	. . . . . . . 8 |- (x e. X -> E.y e. X E.z e. X x = (yGz))
32	31	rgen 1698	. . . . . . 7 |- A.x e. X E.y e. X E.z e. X x = (yGz)
33	27, 1, 32	mpbir2an 730	. . . . . 6 |- G:(X X. X)-onto->X
34		forn 3674	. . . . . 6 |- (G:(X X. X)-onto->X -> ran G = X)
35	33, 34	ax-mp 7	. . . . 5 |- ran G = X
36	35	eqcomi 1479	. . . 4 |- X = ran G
37	36	isgrp 8041	. . 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))))
38	26, 37	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)))
39	22, 38	mpbir 190	1 |- G e. Grp

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646  Vcvv 1811   X. cxp 3168  ran crn 3171  -->wf 3178  -onto->wfo 3180  (class class class)co 3963  Grpcgr 8033

This theorem is referenced by:  issubgi 8122  grpsn 8124  cnaddabl 8126  ablmul 8131  hilabl 9027  symggrpi 10406

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037

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