Theorem abl4pnp 16037

Description: A commutative/associative law for Abelian groups.

Hypotheses

Ref	Expression
abl4pnp.1	|- X = ran G
abl4pnp.2	|- D = ( /g ` G)

Assertion

Ref	Expression
abl4pnp	|- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ F e. X))) -> ((AGB)D(CGF)) = ((ADC)G(BDF)))

Proof of Theorem abl4pnp

Step	Hyp	Ref	Expression
1		abl4pnp.1	. . . . . 6 |- X = ran G
2		abl4pnp.2	. . . . . 6 |- D = ( /g ` G)
3	1, 2	ablmuldiv 9415	. . . . 5 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)DC) = ((ADC)GB))
4		df-3an 860	. . . . 5 |- ((A e. X /\ B e. X /\ C e. X) <-> ((A e. X /\ B e. X) /\ C e. X))
5	3, 4	sylan2br 502	. . . 4 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ C e. X)) -> ((AGB)DC) = ((ADC)GB))
6	5	adantrrr 439	. . 3 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ F e. X))) -> ((AGB)DC) = ((ADC)GB))
7	6	opreq1d 4897	. 2 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ F e. X))) -> (((AGB)DC)DF) = (((ADC)GB)DF))
8		ablgrp 9410	. . . . . . 7 |- (G e. Abel -> G e. Grp)
9	1	grpcl 9324	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ B e. X) -> (AGB) e. X)
10	9	3expib 1070	. . . . . . 7 |- (G e. Grp -> ((A e. X /\ B e. X) -> (AGB) e. X))
11	8, 10	syl 12	. . . . . 6 |- (G e. Abel -> ((A e. X /\ B e. X) -> (AGB) e. X))
12	11	anim1d 619	. . . . 5 |- (G e. Abel -> (((A e. X /\ B e. X) /\ (C e. X /\ F e. X)) -> ((AGB) e. X /\ (C e. X /\ F e. X))))
13		3anass 862	. . . . 5 |- (((AGB) e. X /\ C e. X /\ F e. X) <-> ((AGB) e. X /\ (C e. X /\ F e. X)))
14	12, 13	syl6ibr 230	. . . 4 |- (G e. Abel -> (((A e. X /\ B e. X) /\ (C e. X /\ F e. X)) -> ((AGB) e. X /\ C e. X /\ F e. X)))
15	14	imp 377	. . 3 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ F e. X))) -> ((AGB) e. X /\ C e. X /\ F e. X))
16	1, 2	abldivdiv4 9417	. . 3 |- ((G e. Abel /\ ((AGB) e. X /\ C e. X /\ F e. X)) -> (((AGB)DC)DF) = ((AGB)D(CGF)))
17	15, 16	syldan 516	. 2 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ F e. X))) -> (((AGB)DC)DF) = ((AGB)D(CGF)))
18	1, 2	grpdivcl 9371	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ C e. X) -> (ADC) e. X)
19	18	3expib 1070	. . . . . . 7 |- (G e. Grp -> ((A e. X /\ C e. X) -> (ADC) e. X))
20	19	anim1d 619	. . . . . 6 |- (G e. Grp -> (((A e. X /\ C e. X) /\ (B e. X /\ F e. X)) -> ((ADC) e. X /\ (B e. X /\ F e. X))))
21		an4 564	. . . . . 6 |- (((A e. X /\ B e. X) /\ (C e. X /\ F e. X)) <-> ((A e. X /\ C e. X) /\ (B e. X /\ F e. X)))
22		3anass 862	. . . . . 6 |- (((ADC) e. X /\ B e. X /\ F e. X) <-> ((ADC) e. X /\ (B e. X /\ F e. X)))
23	20, 21, 22	3imtr4g 612	. . . . 5 |- (G e. Grp -> (((A e. X /\ B e. X) /\ (C e. X /\ F e. X)) -> ((ADC) e. X /\ B e. X /\ F e. X)))
24	23	imp 377	. . . 4 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ F e. X))) -> ((ADC) e. X /\ B e. X /\ F e. X))
25	1, 2	grpmuldivass 9373	. . . 4 |- ((G e. Grp /\ ((ADC) e. X /\ B e. X /\ F e. X)) -> (((ADC)GB)DF) = ((ADC)G(BDF)))
26	24, 25	syldan 516	. . 3 |- ((G e. Grp /\ ((A e. X /\ B e. X) /\ (C e. X /\ F e. X))) -> (((ADC)GB)DF) = ((ADC)G(BDF)))
27	26, 8	sylan 497	. 2 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ F e. X))) -> (((ADC)GB)DF) = ((ADC)G(BDF)))
28	7, 17, 27	3eqtr3d 1934	1 |- ((G e. Abel /\ ((A e. X /\ B e. X) /\ (C e. X /\ F e. X))) -> ((AGB)D(CGF)) = ((ADC)G(BDF)))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ran crn 3987  ` cfv 3998  (class class class)co 4884  Grpcgr 9311   /g cgs 9314  Abelcabl 9407

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408

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