Theorem ablo32 27403

Description: Commutative/associative law for Abelian groups. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ablcom.1	|- X = ran G

Assertion

Ref	Expression
ablo32	|- ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( ( A G C ) G B ) )

Proof of Theorem ablo32

Step	Hyp	Ref	Expression
1		ablcom.1	. . . . 5 |- X = ran G
2	1	ablocom 27402	. . . 4 |- ( ( G e. AbelOp /\ B e. X /\ C e. X ) -> ( B G C ) = ( C G B ) )
3	2	3adant3r1 1274	. . 3 |- ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B G C ) = ( C G B ) )
4	3	oveq2d 6666	. 2 |- ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A G ( B G C ) ) = ( A G ( C G B ) ) )
5		ablogrpo 27401	. . 3 |- ( G e. AbelOp -> G e. GrpOp )
6	1	grpoass 27357	. . 3 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( A G ( B G C ) ) )
7	5, 6	sylan 488	. 2 |- ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( A G ( B G C ) ) )
8		3ancomb 1047	. . . 4 |- ( ( A e. X /\ B e. X /\ C e. X ) <-> ( A e. X /\ C e. X /\ B e. X ) )
9	1	grpoass 27357	. . . 4 |- ( ( G e. GrpOp /\ ( A e. X /\ C e. X /\ B e. X ) ) -> ( ( A G C ) G B ) = ( A G ( C G B ) ) )
10	8, 9	sylan2b 492	. . 3 |- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G C ) G B ) = ( A G ( C G B ) ) )
11	5, 10	sylan 488	. 2 |- ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G C ) G B ) = ( A G ( C G B ) ) )
12	4, 7, 11	3eqtr4d 2666	1 |- ( ( G e. AbelOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A G B ) G C ) = ( ( A G C ) G B ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  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-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653  df-grpo 27347  df-ablo 27399

This theorem is referenced by:  ablo4  27404  nvadd32  27478  ip0i  27680  rngoa32  33714