Theorem abldivdiv4 9417

Description: Law for double group division.

Hypotheses

Ref	Expression
abldiv.1	|- X = ran G
abldiv.3	|- D = ( /g ` G)

Assertion

Ref	Expression
abldivdiv4	|- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = (AD(BGC)))

Proof of Theorem abldivdiv4

Step	Hyp	Ref	Expression
1		simpl 346	. . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> G e. Grp)
2		abldiv.1	. . . . . 6 |- X = ran G
3		abldiv.3	. . . . . 6 |- D = ( /g ` G)
4	2, 3	grpdivcl 9371	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) e. X)
5	4	3adant3r3 1079	. . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> (ADB) e. X)
6		simpr3 884	. . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> C e. X)
7		eqid 1884	. . . . 5 |- (inv` G) = (inv` G)
8	2, 7, 3	grpdivval 9367	. . . 4 |- ((G e. Grp /\ (ADB) e. X /\ C e. X) -> ((ADB)DC) = ((ADB)G((inv` G)` C)))
9	1, 5, 6, 8	syl111anc 1100	. . 3 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = ((ADB)G((inv` G)` C)))
10		ablgrp 9410	. . 3 |- (G e. Abel -> G e. Grp)
11	9, 10	sylan 497	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = ((ADB)G((inv` G)` C)))
12		simpr1 882	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> A e. X)
13		simpr2 883	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> B e. X)
14	2, 7	grpinvcl 9352	. . . . 5 |- ((G e. Grp /\ C e. X) -> ((inv` G)` C) e. X)
15		simp3 878	. . . . 5 |- ((A e. X /\ B e. X /\ C e. X) -> C e. X)
16	14, 10, 15	syl2an 503	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((inv` G)` C) e. X)
17	12, 13, 16	3jca 1050	. . 3 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (A e. X /\ B e. X /\ ((inv` G)` C) e. X))
18	2, 3	abldivdiv 9416	. . 3 |- ((G e. Abel /\ (A e. X /\ B e. X /\ ((inv` G)` C) e. X)) -> (AD(BD((inv` G)` C))) = ((ADB)G((inv` G)` C)))
19	17, 18	syldan 516	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (AD(BD((inv` G)` C))) = ((ADB)G((inv` G)` C)))
20	2, 7, 3	grpdivinv 9368	. . . . 5 |- ((G e. Grp /\ B e. X /\ C e. X) -> (BD((inv` G)` C)) = (BGC))
21	20, 10	syl3an1 1130	. . . 4 |- ((G e. Abel /\ B e. X /\ C e. X) -> (BD((inv` G)` C)) = (BGC))
22	21	3adant3r1 1077	. . 3 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (BD((inv` G)` C)) = (BGC))
23	22	opreq2d 4898	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (AD(BD((inv` G)` C))) = (AD(BGC)))
24	11, 19, 23	3eqtr2d 1932	1 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = (AD(BGC)))

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  invcgn 9313   /g cgs 9314  Abelcabl 9407

This theorem is referenced by:  abldiv23 9418  ablnnncan 9419  abl4pnp 16037

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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