Theorem isabl 9409

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 4186	. . . . 5 |- (g = G -> ran g = ran G)
2		isabl.1	. . . . 5 |- X = ran G
3	1, 2	syl6eqr 1946	. . . 4 |- (g = G -> ran g = X)
4		raleq 2266	. . . . 5 |- (ran g = X -> (A.y e. ran g(xgy) = (ygx) <-> A.y e. X (xgy) = (ygx)))
5	4	raleqbi1dv 2271	. . . 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 12	. . 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 4888	. . . . 5 |- (g = G -> (xgy) = (xGy))
8		opreq 4888	. . . . 5 |- (g = G -> (ygx) = (yGx))
9	7, 8	eqeq12d 1899	. . . 4 |- (g = G -> ((xgy) = (ygx) <-> (xGy) = (yGx)))
10	9	2ralbidv 2140	. . 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 587	. 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 9408	. 2 |- Abel = {g e. Grp | A.x e. ran gA.y e. ran g(xgy) = (ygx)}
13	11, 12	elrab2 2416	1 |- (G e. Abel <-> (G e. Grp /\ A.x e. X A.y e. X (xGy) = (yGx)))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  ran crn 3987  (class class class)co 4884  Grpcgr 9311  Abelcabl 9407

This theorem is referenced by:  ablgrp 9410  ablcom 9411  isabli 9414  subgabl 9432  ghgrpi 9445  ablcomgrp 14702  isablda 16035

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-rab 2112  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-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-abl 9408

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