Theorem abl4 8105

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 412	. . . . . . 7 |- (((A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> A e. X)
2	1	adantl 388	. . . . . 6 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> A e. X)
3		simplr 413	. . . . . . 7 |- (((A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> B e. X)
4	3	adantl 388	. . . . . 6 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> B e. X)
5		simprl 414	. . . . . . 7 |- (((A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> C e. X)
6	5	adantl 388	. . . . . 6 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> C e. X)
7	2, 4, 6	3jca 819	. . . . 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 8104	. . . . 5 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)GC) = ((AGC)GB))
10	7, 9	syldan 467	. . . 4 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> ((AGB)GC) = ((AGC)GB))
11	10	opreq1d 3975	. . 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 8044	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AGB) e. X)
13	12	3expb 834	. . . . . . 7 |- ((G e. Grp /\ (A e. X /\ B e. X)) -> (AGB) e. X)
14	13	adantrr 395	. . . . . 6 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (AGB) e. X)
15	5	adantl 388	. . . . . 6 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> C e. X)
16		simprr 415	. . . . . . 7 |- (((A e. X /\ B e. X) /\ (C e. X /\ D e. X)) -> D e. X)
17	16	adantl 388	. . . . . 6 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> D e. X)
18	14, 15, 17	3jca 819	. . . . 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 8047	. . . . 5 |- ((G e. Grp /\ ((AGB) e. X /\ C e. X /\ D e. X)) -> (((AGB)GC)GD) = ((AGB)G(CGD)))
20	18, 19	syldan 467	. . . 4 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (((AGB)GC)GD) = ((AGB)G(CGD)))
21		ablgrp 8102	. . . 4 |- (G e. Abel -> G e. Grp)
22	20, 21	sylan 448	. . 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 8044	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ C e. X) -> (AGC) e. X)
24	23	3expb 834	. . . . . . . 8 |- ((G e. Grp /\ (A e. X /\ C e. X)) -> (AGC) e. X)
25	24	adantrlr 401	. . . . . . 7 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ C e. X)) -> (AGC) e. X)
26	25	adantrrr 403	. . . . . 6 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> (AGC) e. X)
27	3	adantl 388	. . . . . 6 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ D e. X))) -> B e. X)
28	26, 27, 17	3jca 819	. . . . 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 8047	. . . . 5 |- ((G e. Grp /\ ((AGC) e. X /\ B e. X /\ D e. X)) -> (((AGC)GB)GD) = ((AGC)G(BGD)))
30	28, 29	syldan 467	. . . 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 448	. . 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 1515	. 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 829	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 775   = wceq 956   e. wcel 958  ran crn 3171  (class class class)co 3963  Grpcgr 8033  Abelcabl 8099

This theorem is referenced by:  ringa4 8156  vca4 8182  nvadd4 8246  ipdirilem 8488

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-grp 8037  df-abl 8100

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