Theorem abl4 8041

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 817	. . . . 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 8040	. . . . 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 3960	. . 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 7978	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AGB) e. X)
13	12	3expb 832	. . . . . . 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 817	. . . . 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 7981	. . . . 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 8038	. . . 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 7978	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ C e. X) -> (AGC) e. X)
24	23	3expb 832	. . . . . . . 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 817	. . . . 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 7981	. . . . 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 1507	. 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 827	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 773   = wceq 953   e. wcel 955  ran crn 3161  (class class class)co 3948  Grpcgr 7967  Abelcabl 8035

This theorem is referenced by:  ringa4 8093  vca4 8119  nvadd4 8186  ipdirilem 8419

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fo 3186  df-fv 3188  df-opr 3950  df-grp 7971  df-abl 8036

Copyright terms: Public domain