Theorem ablocom 27402

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

Hypothesis

Ref	Expression
ablcom.1	|- X = ran G

Assertion

Ref	Expression
ablocom	|- ( ( G e. AbelOp /\ A e. X /\ B e. X ) -> ( A G B ) = ( B G A ) )

Proof of Theorem ablocom

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablcom.1	. . . . 5 |- X = ran G
2	1	isablo 27400	. . . 4 |- ( G e. AbelOp <-> ( G e. GrpOp /\ A. x e. X A. y e. X ( x G y ) = ( y G x ) ) )
3	2	simprbi 480	. . 3 |- ( G e. AbelOp -> A. x e. X A. y e. X ( x G y ) = ( y G x ) )
4		oveq1 6657	. . . . 5 |- ( x = A -> ( x G y ) = ( A G y ) )
5		oveq2 6658	. . . . 5 |- ( x = A -> ( y G x ) = ( y G A ) )
6	4, 5	eqeq12d 2637	. . . 4 |- ( x = A -> ( ( x G y ) = ( y G x ) <-> ( A G y ) = ( y G A ) ) )
7		oveq2 6658	. . . . 5 |- ( y = B -> ( A G y ) = ( A G B ) )
8		oveq1 6657	. . . . 5 |- ( y = B -> ( y G A ) = ( B G A ) )
9	7, 8	eqeq12d 2637	. . . 4 |- ( y = B -> ( ( A G y ) = ( y G A ) <-> ( A G B ) = ( B G A ) ) )
10	6, 9	rspc2v 3322	. . 3 |- ( ( A e. X /\ B e. X ) -> ( A. x e. X A. y e. X ( x G y ) = ( y G x ) -> ( A G B ) = ( B G A ) ) )
11	3, 10	syl5com 31	. 2 |- ( G e. AbelOp -> ( ( A e. X /\ B e. X ) -> ( A G B ) = ( B G A ) ) )
12	11	3impib 1262	1 |- ( ( G e. AbelOp /\ A e. X /\ B e. X ) -> ( A G B ) = ( B G A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = 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:  ablo32  27403  ablomuldiv  27406  ablodiv32  27409  nvcom  27476  rngocom  33712  iscringd  33797