Theorem isablo 27400

Description: The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
isabl.1	|- X = ran G

Assertion

Ref	Expression
isablo	|- ( G e. AbelOp <-> ( G e. GrpOp /\ A. x e. X A. y e. X ( x G y ) = ( y G x ) ) )

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

Proof of Theorem isablo

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rneq 5351	. . . . 5 |- ( g = G -> ran g = ran G )
2		isabl.1	. . . . 5 |- X = ran G
3	1, 2	syl6eqr 2674	. . . 4 |- ( g = G -> ran g = X )
4		raleq 3138	. . . . 5 |- ( ran g = X -> ( A. y e. ran g ( x g y ) = ( y g x ) <-> A. y e. X ( x g y ) = ( y g x ) ) )
5	4	raleqbi1dv 3146	. . . 4 |- ( ran g = X -> ( A. x e. ran g A. y e. ran g ( x g y ) = ( y g x ) <-> A. x e. X A. y e. X ( x g y ) = ( y g x ) ) )
6	3, 5	syl 17	. . 3 |- ( g = G -> ( A. x e. ran g A. y e. ran g ( x g y ) = ( y g x ) <-> A. x e. X A. y e. X ( x g y ) = ( y g x ) ) )
7		oveq 6656	. . . . 5 |- ( g = G -> ( x g y ) = ( x G y ) )
8		oveq 6656	. . . . 5 |- ( g = G -> ( y g x ) = ( y G x ) )
9	7, 8	eqeq12d 2637	. . . 4 |- ( g = G -> ( ( x g y ) = ( y g x ) <-> ( x G y ) = ( y G x ) ) )
10	9	2ralbidv 2989	. . 3 |- ( g = G -> ( A. x e. X A. y e. X ( x g y ) = ( y g x ) <-> A. x e. X A. y e. X ( x G y ) = ( y G x ) ) )
11	6, 10	bitrd 268	. 2 |- ( g = G -> ( A. x e. ran g A. y e. ran g ( x g y ) = ( y g x ) <-> A. x e. X A. y e. X ( x G y ) = ( y G x ) ) )
12		df-ablo 27399	. 2 |- AbelOp = { g e. GrpOp | A. x e. ran g A. y e. ran g ( x g y ) = ( y g x ) }
13	11, 12	elrab2 3366	1 |- ( G e. AbelOp <-> ( G e. GrpOp /\ A. x e. X A. y e. X ( x G y ) = ( y G x ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   ran crn 5115  (class class class)co 6650   GrpOpcgr 27343   AbelOpcablo 27398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896  df-ov 6653  df-ablo 27399

This theorem is referenced by:  ablogrpo  27401  ablocom  27402  isabloi  27405