Theorem isabl 8097

Description: The predicate "is an Abelian (commutative) group operation."

Hypothesis

Ref	Expression
isabl.1	|- X = ran G

Assertion

Ref	Expression
isabl	|- (G e. Abel <-> (G e. Grp /\ A.x e. X A.y e. X (xGy) = (yGx)))

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

Proof of Theorem isabl

Step	Hyp	Ref	Expression
1		rneq 3345	. . . . 5 |- (g = G -> ran g = ran G)
2		isabl.1	. . . . 5 |- X = ran G
3	1, 2	syl6eqr 1528	. . . 4 |- (g = G -> ran g = X)
4		raleq1 1789	. . . . 5 |- (ran g = X -> (A.y e. ran g(xgy) = (ygx) <-> A.y e. X (xgy) = (ygx)))
5	4	raleqd 1794	. . . 4 |- (ran g = X -> (A.x e. ran gA.y e. ran g(xgy) = (ygx) <-> A.x e. X A.y e. X (xgy) = (ygx)))
6	3, 5	syl 10	. . 3 |- (g = G -> (A.x e. ran gA.y e. ran g(xgy) = (ygx) <-> A.x e. X A.y e. X (xgy) = (ygx)))
7		opreq 3973	. . . . 5 |- (g = G -> (xgy) = (xGy))
8		opreq 3973	. . . . 5 |- (g = G -> (ygx) = (yGx))
9	7, 8	eqeq12d 1492	. . . 4 |- (g = G -> ((xgy) = (ygx) <-> (xGy) = (yGx)))
10	9	2ralbidv 1683	. . 3 |- (g = G -> (A.x e. X A.y e. X (xgy) = (ygx) <-> A.x e. X A.y e. X (xGy) = (yGx)))
11	6, 10	bitrd 530	. 2 |- (g = G -> (A.x e. ran gA.y e. ran g(xgy) = (ygx) <-> A.x e. X A.y e. X (xGy) = (yGx)))
12		df-abl 8096	. 2 |- Abel = {g e. Grp | A.x e. ran gA.y e. ran g(xgy) = (ygx)}
13	11, 12	elrab2 1910	1 |- (G e. Abel <-> (G e. Grp /\ A.x e. X A.y e. X (xGy) = (yGx)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  ran crn 3177  (class class class)co 3969  Grpcgr 8030  Abelcabl 8095

This theorem is referenced by:  ablgrp 8098  ablcom 8099  isabli 8102  subgabl 8119  ghgrpi 8133

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971  df-abl 8096

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