Theorem grpdivval 8078

Description: Group division (or subtraction) operation value.

Hypotheses

Ref	Expression
grpdiv.1	|- X = ran G
grpdiv.2	|- N = (inv` G)
grpdiv.3	|- D = ( /g ` G)

Assertion

Ref	Expression
grpdivval	|- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) = (AG(N` B)))

Proof of Theorem grpdivval

Step	Hyp	Ref	Expression
1		grpdiv.1	. . . . 5 |- X = ran G
2		grpdiv.2	. . . . 5 |- N = (inv` G)
3		grpdiv.3	. . . . 5 |- D = ( /g ` G)
4	1, 2, 3	grpdivfval 8077	. . . 4 |- (G e. Grp -> D = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))})
5	4	opreqd 3983	. . 3 |- (G e. Grp -> (ADB) = (A{<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))}B))
6	5	3ad2ant1 802	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) = (A{<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))}B))
7		oprex 3989	. . . 4 |- (AG(N` B)) e. V
8		opreq1 3974	. . . 4 |- (x = A -> (xG(N` y)) = (AG(N` y)))
9		fveq2 3730	. . . . 5 |- (y = B -> (N` y) = (N` B))
10	9	opreq2d 3982	. . . 4 |- (y = B -> (AG(N` y)) = (AG(N` B)))
11		eqid 1478	. . . 4 |- {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))} = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))}
12	7, 8, 10, 11	oprabval2 4034	. . 3 |- ((A e. X /\ B e. X) -> (A{<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))}B) = (AG(N` B)))
13	12	3adant1 799	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> (A{<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))}B) = (AG(N` B)))
14	6, 13	eqtrd 1510	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> (ADB) = (AG(N` B)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  ran crn 3177  ` cfv 3188  (class class class)co 3969  {copab2 3970  Grpcgr 8030  invcgn 8032   /g cgs 8033

This theorem is referenced by:  grpdivinv 8079  grpinvdiv 8080  grpdivdiv 8083  grpmuldivass 8084  grpdivid 8085  grpnpcan 8087  grppnpcan2 8088  grpnnncan2 8089  abldivdiv4 8105  nvmval 8259

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-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-rex 1653  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-fv 3204  df-opr 3971  df-oprab 3972  df-gdiv 8037

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