Theorem grplactval 8093

Description: The value of the left group action of element A of group G at B. (Contributed by Paul Chapman, 18-Mar-2008.)

Hypotheses

Ref	Expression
grplact.1	|- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
grplact.2	|- X = ran G

Assertion

Ref	Expression
grplactval	|- ((G e. Grp /\ A e. X /\ B e. X) -> ((F` A)` B) = (AGB))

Distinct variable groups:   A,a,b,g,h   B,a,b   G,a,b,g,h   X,a,b,g,h

Proof of Theorem grplactval

Step	Hyp	Ref	Expression
1		grplact.1	. . . . 5 |- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
2		grplact.2	. . . . 5 |- X = ran G
3	1, 2	grplactfval 8092	. . . 4 |- ((G e. Grp /\ A e. X) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})
4	3	3adant3 801	. . 3 |- ((G e. Grp /\ A e. X /\ B e. X) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})
5	4	fveq1d 3732	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((F` A)` B) = ({<.a, b>. | (a e. X /\ b = (AGa))}` B))
6		opreq2 3975	. . . 4 |- (a = B -> (AGa) = (AGB))
7		eqid 1478	. . . 4 |- {<.a, b>. | (a e. X /\ b = (AGa))} = {<.a, b>. | (a e. X /\ b = (AGa))}
8		oprex 3989	. . . 4 |- (AGB) e. V
9	6, 7, 8	fvopab4 3786	. . 3 |- (B e. X -> ({<.a, b>. | (a e. X /\ b = (AGa))}` B) = (AGB))
10	9	3ad2ant3 804	. 2 |- ((G e. Grp /\ A e. X /\ B e. X) -> ({<.a, b>. | (a e. X /\ b = (AGa))}` B) = (AGB))
11	5, 10	eqtrd 1510	1 |- ((G e. Grp /\ A e. X /\ B e. X) -> ((F` A)` B) = (AGB))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  {copab 2671  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030

This theorem is referenced by:  grplactf1o 8094  shftefif1olem 8736  cayleylem2 10405  cayleylem3 10406

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-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

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