Theorem abl4 8101

Description: Commutative/associative law for Abelian groups.

Hypothesis

Ref	Expression
ablcom.1	|- X = ran G

Assertion

Ref	Expression
abl4	|- ((G e. Abel /\ (A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> ((AGB)G(CGD)) = ((AGC)G(BGD)))

Proof of Theorem abl4

Step	Hyp	Ref	Expression
1		simpll 414	. . . . . . 7 |- (((A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> A e. X)
2	1	adantl 390	. . . . . 6 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> A e. X)
3		simplr 415	. . . . . . 7 |- (((A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> B e. X)
4	3	adantl 390	. . . . . 6 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> B e. X)
5		simprl 416	. . . . . . 7 |- (((A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> C e. X)
6	5	adantl 390	. . . . . 6 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> C e. X)
7	2, 4, 6	3jca 821	. . . . 5 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (A e. X /\ B e. X /\ C e. X))
8		ablcom.1	. . . . . 6 |- X = ran G
9	8	abl23 8100	. . . . 5 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = ((AGC)GB))
10	7, 9	syldan 469	. . . 4 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> ((AGB)GC) = ((AGC)GB))
11	10	opreq1d 3981	. . 3 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (((AGB)GC)GD) = (((AGC)GB)GD))
12	8	grpcl 8041	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AGB) e. X)
13	12	3expb 836	. . . . . . 7 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (AGB) e. X)
14	13	adantrr 397	. . . . . 6 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (AGB) e. X)
15	5	adantl 390	. . . . . 6 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> C e. X)
16		simprr 417	. . . . . . 7 |- (((A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> D e. X)
17	16	adantl 390	. . . . . 6 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> D e. X)
18	14, 15, 17	3jca 821	. . . . 5 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> ((AGB) e. X /\ C e. X /\ D e. X))
19	8	grpass 8044	. . . . 5 |- ((G e. Grp /\ ((AGB) e. X /\ C e. X /\ D e. X)) -> (((AGB)GC)GD) = ((AGB)G(CGD)))
20	18, 19	syldan 469	. . . 4 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (((AGB)GC)GD) = ((AGB)G(CGD)))
21		ablgrp 8098	. . . 4 |- (G e. Abel -> G e. Grp)
22	20, 21	sylan 450	. . 3 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (((AGB)GC)GD) = ((AGB)G(CGD)))
23	8	grpcl 8041	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ C e. X) -> (AGC) e. X)
24	23	3expb 836	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> (AGC) e. X)
25	24	adantrlr 403	. . . . . . 7 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ C e. X)) -> (AGC) e. X)
26	25	adantrrr 405	. . . . . 6 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (AGC) e. X)
27	3	adantl 390	. . . . . 6 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> B e. X)
28	26, 27, 17	3jca 821	. . . . 5 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> ((AGC) e. X /\ B e. X /\ D e. X))
29	8	grpass 8044	. . . . 5 |- ((G e. Grp /\ ((AGC) e. X /\ B e. X /\ D e. X)) -> (((AGC)GB)GD) = ((AGC)G(BGD)))
30	28, 29	syldan 469	. . . 4 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (((AGC)GB)GD) = ((AGC)G(BGD)))
31	30, 21	sylan 450	. . 3 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (((AGC)GB)GD) = ((AGC)G(BGD)))
32	11, 22, 31	3eqtr3d 1518	. 2 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> ((AGB)G(CGD)) = ((AGC)G(BGD)))
33	32	3impb 831	1 |- ((G e. Abel /\ (A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> ((AGB)G(CGD)) = ((AGC)G(BGD)))

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:  ringa4 8152  vca4 8178  nvadd4 8242  ipdirilem 8484

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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