Theorem abldivdiv4 8105

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		abldiv.1	. . . . 5 |- X = ran G
2		eqid 1478	. . . . 5 |- (inv` G) = (inv` G)
3		abldiv.3	. . . . 5 |- D = ( /g ` G)
4	1, 2, 3	grpdivval 8078	. . . 4 |- ((G e. Grp /\ (ADB) e. X /\ C e. X) -> ((ADB)DC) = ((ADB)G((inv` G)` C)))
5		pm3.26 319	. . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> G e. Grp)
6	1, 3	grpdivcl 8082	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) e. X)
7	6	3adant3r3 846	. . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> (ADB) e. X)
8		3simp3 792	. . . . 5 |- ((A e. X /\ B e. X /\ C e. X) -> C e. X)
9	8	adantl 390	. . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> C e. X)
10	4, 5, 7, 9	syl3anc 860	. . 3 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = ((ADB)G((inv` G)` C)))
11		ablgrp 8098	. . 3 |- (G e. Abel -> G e. Grp)
12	10, 11	sylan 450	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = ((ADB)G((inv` G)` C)))
13		3simp1 790	. . . . 5 |- ((A e. X /\ B e. X /\ C e. X) -> A e. X)
14	13	adantl 390	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> A e. X)
15		3simp2 791	. . . . 5 |- ((A e. X /\ B e. X /\ C e. X) -> B e. X)
16	15	adantl 390	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> B e. X)
17	1, 2	grpinvcl 8064	. . . . 5 |- ((G e. Grp /\ C e. X) -> ((inv` G)` C) e. X)
18	17, 11, 8	syl2an 456	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((inv` G)` C) e. X)
19	14, 16, 18	3jca 821	. . 3 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (A e. X /\ B e. X /\ ((inv` G)` C) e. X))
20	1, 3	abldivdiv 8104	. . 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)))
21	19, 20	syldan 469	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (AD(BD((inv` G)` C))) = ((ADB)G((inv` G)` C)))
22	1, 2, 3	grpdivinv 8079	. . . . 5 |- ((G e. Grp /\ B e. X /\ C e. X) -> (BD((inv` G)` C)) = (BGC))
23	22, 11	syl3an1 861	. . . 4 |- ((G e. Abel /\ B e. X /\ C e. X) -> (BD((inv` G)` C)) = (BGC))
24	23	3adant3r1 844	. . 3 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (BD((inv` G)` C)) = (BGC))
25	24	opreq2d 3982	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (AD(BD((inv` G)` C))) = (AD(BGC)))
26	12, 21, 25	3eqtr2d 1516	1 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = (AD(BGC)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  invcgn 8032   /g cgs 8033  Abelcabl 8095

This theorem is referenced by:  abldiv23 8106  ablnnncan 8107

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-9 967  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-rep 2698  ax-sep 2708  ax-nul 2715  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-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  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-oprab 3972  df-1st 4085  df-2nd 4086  df-grp 8034  df-gid 8035  df-ginv 8036  df-gdiv 8037  df-abl 8096

Copyright terms: Public domain