Theorem isabl 8101

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 3339	. . . . 5 |- (g = G -> ran g = ran G)
2		isabl.1	. . . . 5 |- X = ran G
3	1, 2	syl6eqr 1525	. . . 4 |- (g = G -> ran g = X)
4		raleq1 1786	. . . . 5 |- (ran g = X -> (A.y e. ran g(xgy) = (ygx) <-> A.y e. X (xgy) = (ygx)))
5	4	raleqd 1791	. . . 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 3967	. . . . 5 |- (g = G -> (xgy) = (xGy))
8		opreq 3967	. . . . 5 |- (g = G -> (ygx) = (yGx))
9	7, 8	eqeq12d 1489	. . . 4 |- (g = G -> ((xgy) = (ygx) <-> (xGy) = (yGx)))
10	9	2ralbidv 1680	. . 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 528	. 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 8100	. 2 |- Abel = {g e. Grp | A.x e. ran gA.y e. ran g(xgy) = (ygx)}
13	11, 12	elrab2 1907	1 |- (G e. Abel <-> (G e. Grp /\ A.x e. X A.y e. X (xGy) = (yGx)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  ran crn 3171  (class class class)co 3963  Grpcgr 8033  Abelcabl 8099

This theorem is referenced by:  ablgrp 8102  ablcom 8103  isabli 8106  subgabl 8123  ghgrpi 8137

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-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-rab 1652  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-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965  df-abl 8100

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