Theorem ablmuldiv 8103

Description: Law for group multiplication and division.

Hypotheses

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

Assertion

Ref	Expression
ablmuldiv	|- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)DC) = ((ADC)GB))

Proof of Theorem ablmuldiv

Step	Hyp	Ref	Expression
1		abldiv.1	. . . . 5 |- X = ran G
2	1	ablcom 8099	. . . 4 |- ((G e. Abel /\ A e. X /\ B e. X) -> (AGB) = (BGA))
3	2	3adant3r3 846	. . 3 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (AGB) = (BGA))
4	3	opreq1d 3981	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)DC) = ((BGA)DC))
5		abldiv.3	. . . . 5 |- D = ( /g ` G)
6	1, 5	grpmuldivass 8084	. . . 4 |- ((G e. Grp /\ (B e. X /\ A e. X /\ C e. X)) -> ((BGA)DC) = (BG(ADC)))
7		ablgrp 8098	. . . 4 |- (G e. Abel -> G e. Grp)
8	6, 7	sylan 450	. . 3 |- ((G e. Abel /\ (B e. X /\ A e. X /\ C e. X)) -> ((BGA)DC) = (BG(ADC)))
9		3ancoma 784	. . 3 |- ((A e. X /\ B e. X /\ C e. X) <-> (B e. X /\ A e. X /\ C e. X))
10	8, 9	sylan2b 454	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((BGA)DC) = (BG(ADC)))
11		3simp2 791	. . . . 5 |- ((A e. X /\ B e. X /\ C e. X) -> B e. X)
12	11	adantl 390	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> B e. X)
13	1, 5	grpdivcl 8082	. . . . . 6 |- ((G e. Grp /\ A e. X /\ C e. X) -> (ADC) e. X)
14	13, 7	syl3an1 861	. . . . 5 |- ((G e. Abel /\ A e. X /\ C e. X) -> (ADC) e. X)
15	14	3adant3r2 845	. . . 4 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (ADC) e. X)
16	12, 15	jca 288	. . 3 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (B e. X /\ (ADC) e. X))
17	1	ablcom 8099	. . . 4 |- ((G e. Abel /\ B e. X /\ (ADC) e. X) -> (BG(ADC)) = ((ADC)GB))
18	17	3expb 836	. . 3 |- ((G e. Abel /\ (B e. X /\ (ADC) e. X)) -> (BG(ADC)) = ((ADC)GB))
19	16, 18	syldan 469	. 2 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> (BG(ADC)) = ((ADC)GB))
20	4, 10, 19	3eqtrd 1514	1 |- ((G e. Abel /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)DC) = ((ADC)GB))

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   /g cgs 8033  Abelcabl 8095

This theorem is referenced by:  abldivdiv 8104  nvaddsub 8275

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

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