Theorem abl23 8100

Description: Commutative/associative law for Abelian groups.

Hypothesis

Ref	Expression
ablcom.1	|- X = ran G

Assertion

Ref	Expression
abl23	|- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = ((AGC)GB))

Proof of Theorem abl23

Step	Hyp	Ref	Expression
1		ablcom.1	. . . . 5 |- X = ran G
2	1	ablcom 8099	. . . 4 |- ((G e. Abel /\ B e. X /\ C e. X) -> (BGC) = (CGB))
3	2	3adant3r1 844	. . 3 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (BGC) = (CGB))
4	3	opreq2d 3982	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (AG(BGC)) = (AG(CGB)))
5	1	grpass 8044	. . 3 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = (AG(BGC)))
6		ablgrp 8098	. . 3 |- (G e. Abel -> G e. Grp)
7	5, 6	sylan 450	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = (AG(BGC)))
8	1	grpass 8044	. . . 4 |- ((G e. Grp /\ (A e. X /\ C e. X /\ B e. X)) -> ((AGC)GB) = (AG(CGB)))
9		3ancomb 785	. . . 4 |- ((A e. X /\ B e. X /\ C e. X) <-> (A e. X /\ C e. X /\ B e. X))
10	8, 9	sylan2b 454	. . 3 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGC)GB) = (AG(CGB)))
11	10, 6	sylan 450	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGC)GB) = (AG(CGB)))
12	4, 7, 11	3eqtr4d 1520	1 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = ((AGC)GB))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  ran crn 3177  (class class class)co 3969  Grpcgr 8030  Abelcabl 8095

This theorem is referenced by:  abl4 8101  ringa23 8151  vca23 8177  nvadd23 8239  ip0i 8480

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-grp 8034  df-abl 8096

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