Theorem abldivdiv4 8109

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 1475	. . . . 5 |- (inv` G) = (inv` G)
3		abldiv.3	. . . . 5 |- D = ( /g ` G)
4	1, 2, 3	grpdivval 8082	. . . 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 8086	. . . . 5 |- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) e. X)
7	6	3adant3r3 844	. . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> (ADB) e. X)
8		3simp3 790	. . . . 5 |- ((A e. X /\ B e. X /\ C e. X) -> C e. X)
9	8	adantl 388	. . . 4 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> C e. X)
10	4, 5, 7, 9	syl3anc 858	. . 3 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = ((ADB)G((inv` G)` C)))
11		ablgrp 8102	. . 3 |- (G e. Abel -> G e. Grp)
12	10, 11	sylan 448	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = ((ADB)G((inv` G)` C)))
13		3simp1 788	. . . . 5 |- ((A e. X /\ B e. X /\ C e. X) -> A e. X)
14	13	adantl 388	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> A e. X)
15		3simp2 789	. . . . 5 |- ((A e. X /\ B e. X /\ C e. X) -> B e. X)
16	15	adantl 388	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> B e. X)
17	1, 2	grpinvcl 8068	. . . . 5 |- ((G e. Grp /\ C e. X) -> ((inv` G)` C) e. X)
18	17, 11, 8	syl2an 454	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((inv` G)` C) e. X)
19	14, 16, 18	3jca 819	. . 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 8108	. . 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 467	. 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 8083	. . . . 5 |- ((G e. Grp /\ B e. X /\ C e. X) -> (BD((inv` G)` C)) = (BGC))
23	22, 11	syl3an1 859	. . . 4 |- ((G e. Abel /\ B e. X /\ C e. X) -> (BD((inv` G)` C)) = (BGC))
24	23	3adant3r1 842	. . 3 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (BD((inv` G)` C)) = (BGC))
25	24	opreq2d 3976	. 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 1513	1 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADB)DC) = (AD(BGC)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  invcgn 8035   /g cgs 8036  Abelcabl 8099

This theorem is referenced by:  abldiv23 8110  ablnnncan 8111

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-9 965  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-rep 2693  ax-sep 2703  ax-nul 2710  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-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  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-oprab 3966  df-1st 4079  df-2nd 4080  df-grp 8037  df-gid 8038  df-ginv 8039  df-gdiv 8040  df-abl 8100

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