Theorem grpnnncan2 8076

Description: Group theory analog of nnncan2t 5455.

Hypotheses

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

Assertion

Ref	Expression
grpnnncan2	|- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADC)D(BDC)) = (ADB))

Proof of Theorem grpnnncan2

Step	Hyp	Ref	Expression
1		grpdivf.1	. . . . 5 |- X = ran G
2		eqid 1475	. . . . 5 |- (inv` G) = (inv` G)
3		grpdivf.3	. . . . 5 |- D = ( /g ` G)
4	1, 2, 3	grpdivval 8065	. . . 4 |- ((G e. Grp /\ A e. X /\ C e. X) -> (ADC) = (AG((inv` G)` C)))
5	4	3adant3r2 842	. . 3 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> (ADC) = (AG((inv` G)` C)))
6	1, 2, 3	grpdivval 8065	. . . 4 |- ((G e. Grp /\ B e. X /\ C e. X) -> (BDC) = (BG((inv` G)` C)))
7	6	3adant3r1 841	. . 3 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> (BDC) = (BG((inv` G)` C)))
8	5, 7	opreq12d 3975	. 2 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADC)D(BDC)) = ((AG((inv` G)` C))D(BG((inv` G)` C))))
9		idd 61	. . . . 5 |- (G e. Grp -> (A e. X -> A e. X))
10		idd 61	. . . . 5 |- (G e. Grp -> (B e. X -> B e. X))
11	1, 2	grpinvcl 8051	. . . . . 6 |- ((G e. Grp /\ C e. X) -> ((inv` G)` C) e. X)
12	11	ex 373	. . . . 5 |- (G e. Grp -> (C e. X -> ((inv` G)` C) e. X))
13	9, 10, 12	3anim123d 899	. . . 4 |- (G e. Grp -> ((A e. X /\ B e. X /\ C e. X) -> (A e. X /\ B e. X /\ ((inv` G)` C) e. X)))
14	13	imp 350	. . 3 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> (A e. X /\ B e. X /\ ((inv` G)` C) e. X))
15	1, 3	grppnpcan2 8075	. . 3 |- ((G e. Grp /\ (A e. X /\ B e. X /\ ((inv` G)` C) e. X)) -> ((AG((inv` G)` C))D(BG((inv` G)` C))) = (ADB))
16	14, 15	syldan 467	. 2 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((AG((inv` G)` C))D(BG((inv` G)` C))) = (ADB))
17	8, 16	eqtrd 1506	1 |- ((G e. Grp /\ (A e. X /\ B e. X /\ C e. X)) -> ((ADC)D(BDC)) = (ADB))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  ran crn 3168  ` cfv 3179  (class class class)co 3960  Grpcgr 8016  invcgn 8018   /g cgs 8019

This theorem is referenced by:  nvnnncan2 8254

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fo 3193  df-fv 3195  df-opr 3962  df-oprab 3963  df-grp 8020  df-gid 8021  df-ginv 8022  df-gdiv 8023

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