Theorem isgrpi 7992

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 1721	. . 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 3959	. . . . . . . . 9 |- (y = N -> (yGx) = (NGx))
7	6	eqeq1d 1480	. . . . . . . 8 |- (y = N -> ((yGx) = U <-> (NGx) = U))
8	7	rcla4ev 1873	. . . . . . 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 1695	. . . 4 |- A.x e. X ((UGx) = x /\ E.y e. X (yGx) = U)
14		opreq1 3959	. . . . . . . 8 |- (u = U -> (uGx) = (UGx))
15	14	eqeq1d 1480	. . . . . . 7 |- (u = U -> ((uGx) = x <-> (UGx) = x))
16		eqeq2 1481	. . . . . . . 8 |- (u = U -> ((yGx) = u <-> (yGx) = U))
17	16	rexbidv 1661	. . . . . . 7 |- (u = U -> (E.y e. X (yGx) = u <-> E.y e. X (yGx) = U))
18	15, 17	anbi12d 627	. . . . . 6 |- (u = U -> (((uGx) = x /\ E.y e. X (yGx) = u) <-> ((UGx) = x /\ E.y e. X (yGx) = U)))
19	18	ralbidv 1660	. . . . 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 1873	. . . 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 696	. . 3 |- E.u e. X A.x e. X ((uGx) = x /\ E.y e. X (yGx) = u)
22	1, 3, 21	3pm3.2i 817	. 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 3255	. . . 4 |- (X X. X) e. V
25		fex 3643	. . . 4 |- ((G:(X X. X)-->X /\ (X X. X) e. V) -> G e. V)
26	1, 24, 25	mp2an 696	. . 3 |- G e. V
27		fooprval 4028	. . . . . . 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 1477	. . . . . . . . 9 |- (x e. X -> x = (UGx))
29		rcla4eopr 3981	. . . . . . . . . 10 |- ((U e. X /\ x e. X /\ x = (UGx)) -> E.y e. X E.z e. X x = (yGz))
30	4, 29	mp3an1 901	. . . . . . . . 9 |- ((x e. X /\ x = (UGx)) -> E.y e. X E.z e. X x = (yGz))
31	28, 30	mpdan 703	. . . . . . . 8 |- (x e. X -> E.y e. X E.z e. X x = (yGz))
32	31	rgen 1695	. . . . . . 7 |- A.x e. X E.y e. X E.z e. X x = (yGz)
33	27, 1, 32	mpbir2an 729	. . . . . 6 |- G:(X X. X)-onto->X
34		forn 3665	. . . . . 6 |- (G:(X X. X)-onto->X -> ran G = X)
35	33, 34	ax-mp 7	. . . . 5 |- ran G = X
36	35	eqcomi 1476	. . . 4 |- X = ran G
37	36	isgrp 7991	. . 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 774   = wceq 954   e. wcel 956  A.wral 1642  E.wrex 1643  Vcvv 1807   X. cxp 3163  ran crn 3166  -->wf 3173  -onto->wfo 3175  (class class class)co 3954  Grpcgr 7983

This theorem is referenced by:  issubgi 8074  grpsn 8076  cnaddabl 8078  ablmul 8083  circgrpOLD 8677  hilabl 8966  symggrpi 10340

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191  df-fv 3193  df-opr 3956  df-grp 7987

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