Theorem grpdivfval 8077

Description: Group division (or subtraction) operation.

Hypotheses

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

Assertion

Ref	Expression
grpdivfval	|- (G e. Grp -> D = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))})

Distinct variable groups:   x,y,z,G   x,N,y,z   x,X,y,z

Proof of Theorem grpdivfval

Step	Hyp	Ref	Expression
1		eqid 1478	. . . . 5 |- {<.<.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)))}
2	1	oprabex2g 4026	. . . 4 |- ((X e. V /\ X e. V) -> {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))} e. V)
3		rnexg 3365	. . . . 5 |- (G e. Grp -> ran G e. V)
4		grpdiv.1	. . . . 5 |- X = ran G
5	3, 4	syl5eqel 1555	. . . 4 |- (G e. Grp -> X e. V)
6	2, 5, 5	sylanc 473	. . 3 |- (G e. Grp -> {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))} e. V)
7		rneq 3345	. . . . . . . . 9 |- (g = G -> ran g = ran G)
8	7, 4	syl6eqr 1528	. . . . . . . 8 |- (g = G -> ran g = X)
9	8	eleq2d 1544	. . . . . . 7 |- (g = G -> (x e. ran g <-> x e. X))
10	8	eleq2d 1544	. . . . . . 7 |- (g = G -> (y e. ran g <-> y e. X))
11	9, 10	anbi12d 630	. . . . . 6 |- (g = G -> ((x e. ran g /\ y e. ran g) <-> (x e. X /\ y e. X)))
12		opreq 3973	. . . . . . . 8 |- (g = G -> (xg((inv` g)` y)) = (xG((inv` g)` y)))
13		fveq2 3730	. . . . . . . . . . 11 |- (g = G -> (inv` g) = (inv` G))
14		grpdiv.2	. . . . . . . . . . 11 |- N = (inv` G)
15	13, 14	syl6eqr 1528	. . . . . . . . . 10 |- (g = G -> (inv` g) = N)
16	15	fveq1d 3732	. . . . . . . . 9 |- (g = G -> ((inv` g)` y) = (N` y))
17	16	opreq2d 3982	. . . . . . . 8 |- (g = G -> (xG((inv` g)` y)) = (xG(N` y)))
18	12, 17	eqtrd 1510	. . . . . . 7 |- (g = G -> (xg((inv` g)` y)) = (xG(N` y)))
19	18	eqeq2d 1489	. . . . . 6 |- (g = G -> (z = (xg((inv` g)` y)) <-> z = (xG(N` y))))
20	11, 19	anbi12d 630	. . . . 5 |- (g = G -> (((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y))) <-> ((x e. X /\ y e. X) /\ z = (xG(N` y)))))
21	20	oprabbidv 4002	. . . 4 |- (g = G -> {<.<.x, y>., z>. | ((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y)))} = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))})
22		df-gdiv 8037	. . . 4 |- /g = {<.g, f>. | (g e. Grp /\ f = {<.<.x, y>., z>. | ((x e. ran g /\ y e. ran g) /\ z = (xg((inv` g)` y)))})}
23	21, 22	fvopab4g 3785	. . 3 |- ((G e. Grp /\ {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))} e. V) -> ( /g ` G) = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))})
24	6, 23	mpdan 706	. 2 |- (G e. Grp -> ( /g ` G) = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))})
25		grpdiv.3	. 2 |- D = ( /g ` G)
26	24, 25	syl5eq 1522	1 |- (G e. Grp -> D = {<.<.x, y>., z>. | ((x e. X /\ y e. X) /\ z = (xG(N` y)))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  ran crn 3177  ` cfv 3188  (class class class)co 3969  {copab2 3970  Grpcgr 8030  invcgn 8032   /g cgs 8033

This theorem is referenced by:  grpdivval 8078  grpdivf 8081  nvmfval 8260

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-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-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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