Theorem isga 9450

Description: The predicate "is a (left) group action." The group G is said to act on the base set Y of the action, which is not assumed to have any special properties. There is a related notion of right group action, but as the Wikipedia article explains, it is not mathematically interesting. The way actions are usually thought of is that each element g of G is a permutation of the elements of Y (see gapm 9462). Since group theory was classically about symmetry groups, it is therefore likely that the notion of group action was useful even in early group theory. (Contributed by Jeff Hankins, 10-Aug-2009.)

Hypotheses

Ref	Expression
isga.1	|- X = ran G
isga.2	|- Y = ran dom M
isga.3	|- U = (Id` G)

Assertion

Ref	Expression
isga	|- (M e. A -> (<.G, M>. e. GrpAct <-> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))

Distinct variable groups:   x,y,z,G   x,M,y,z   y,X,z   x,Y

Proof of Theorem isga

Step	Hyp	Ref	Expression
1		isga.1	. . 3 |- X = ran G
2		isga.3	. . 3 |- U = (Id` G)
3	1, 2	isgalem 9449	. 2 |- (M e. A -> (<.G, M>. e. GrpAct <-> (G e. Grp /\ E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))))
4		fdm 4567	. . . . . . . . . . . . 13 |- (M:(X X. s)-->s -> dom M = (X X. s))
5	4	adantr 425	. . . . . . . . . . . 12 |- ((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> dom M = (X X. s))
6	5	3ad2ant1 897	. . . . . . . . . . 11 |- (((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) /\ G e. Grp /\ M e. A) -> dom M = (X X. s))
7	6	rneqd 4188	. . . . . . . . . 10 |- (((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) /\ G e. Grp /\ M e. A) -> ran dom M = ran ( X X. s))
8	1	grpn0 9326	. . . . . . . . . . . 12 |- (G e. Grp -> X =/= (/))
9	8	3ad2ant2 898	. . . . . . . . . . 11 |- (((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) /\ G e. Grp /\ M e. A) -> X =/= (/))
10		rnxp 4342	. . . . . . . . . . 11 |- (X =/= (/) -> ran ( X X. s) = s)
11	9, 10	syl 12	. . . . . . . . . 10 |- (((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) /\ G e. Grp /\ M e. A) -> ran ( X X. s) = s)
12	7, 11	eqtr2d 1926	. . . . . . . . 9 |- (((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) /\ G e. Grp /\ M e. A) -> s = ran dom M)
13		isga.2	. . . . . . . . 9 |- Y = ran dom M
14	12, 13	syl6eqr 1946	. . . . . . . 8 |- (((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) /\ G e. Grp /\ M e. A) -> s = Y)
15		xpeq2 4017	. . . . . . . . . . . . . 14 |- (s = Y -> (X X. s) = (X X. Y))
16		feq23 4554	. . . . . . . . . . . . . 14 |- (((X X. s) = (X X. Y) /\ s = Y) -> (M:(X X. s)-->s <-> M:(X X. Y)-->Y))
17	15, 16	mpancom 769	. . . . . . . . . . . . 13 |- (s = Y -> (M:(X X. s)-->s <-> M:(X X. Y)-->Y))
18		raleq 2266	. . . . . . . . . . . . 13 |- (s = Y -> (A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) <-> A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))
19	17, 18	3anbi23d 1171	. . . . . . . . . . . 12 |- (s = Y -> ((G e. Grp /\ M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) <-> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
20	19	biimpcd 172	. . . . . . . . . . 11 |- ((G e. Grp /\ M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> (s = Y -> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
21	20	3expb 1068	. . . . . . . . . 10 |- ((G e. Grp /\ (M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))) -> (s = Y -> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
22	21	ancoms 484	. . . . . . . . 9 |- (((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) /\ G e. Grp) -> (s = Y -> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
23	22	3adant3 896	. . . . . . . 8 |- (((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) /\ G e. Grp /\ M e. A) -> (s = Y -> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
24	14, 23	mpd 29	. . . . . . 7 |- (((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) /\ G e. Grp /\ M e. A) -> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))
25	24	3exp 1066	. . . . . 6 |- ((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> (G e. Grp -> (M e. A -> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))))
26	25	19.23aiv 1674	. . . . 5 |- (E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> (G e. Grp -> (M e. A -> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))))
27	26	impcom 378	. . . 4 |- ((G e. Grp /\ E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))) -> (M e. A -> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
28	27	com12 14	. . 3 |- (M e. A -> ((G e. Grp /\ E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))) -> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
29		dmexg 4206	. . . . . . . 8 |- (M e. A -> dom M e. _V)
30		rnexg 4207	. . . . . . . . 9 |- (dom M e. _V -> ran dom M e. _V)
31	30, 13	syl5eqel 1975	. . . . . . . 8 |- (dom M e. _V -> Y e. _V)
32	17, 18	anbi12d 690	. . . . . . . . 9 |- (s = Y -> ((M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) <-> (M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
33	32	cla4egv 2365	. . . . . . . 8 |- (Y e. _V -> ((M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
34	29, 31, 33	3syl 24	. . . . . . 7 |- (M e. A -> ((M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
35	34	anim2d 620	. . . . . 6 |- (M e. A -> ((G e. Grp /\ (M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))) -> (G e. Grp /\ E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))))
36	35	com12 14	. . . . 5 |- ((G e. Grp /\ (M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))) -> (M e. A -> (G e. Grp /\ E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))))
37	36	3impb 1063	. . . 4 |- ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> (M e. A -> (G e. Grp /\ E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))))
38	37	com12 14	. . 3 |- (M e. A -> ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> (G e. Grp /\ E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))))
39	28, 38	impbid 574	. 2 |- (M e. A -> ((G e. Grp /\ E.s(M:(X X. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))) <-> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
40	3, 39	bitrd 587	1 |- (M e. A -> (<.G, M>. e. GrpAct <-> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  _Vcvv 2292  (/)c0 2875  <.cop 3046   X. cxp 3984  dom cdm 3986  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  GrpActcga 9447

This theorem is referenced by:  gafo 9451  isga2 9452  ltlga 14729

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-grp 9316  df-ga 9448

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