Theorem isgalem 9449

Description: Lemma for isga 9450.

Hypotheses

Ref	Expression
isgalem.1	|- X = ran G
isgalem.2	|- U = (Id` G)

Assertion

Ref	Expression
isgalem	|- (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)))))))

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

Proof of Theorem isgalem

Step	Hyp	Ref	Expression
1		df-br 3339	. . . . 5 |- (GGrpActM <-> <.G, M>. e. GrpAct)
2		relopab 4104	. . . . . . 7 |- Rel {<.g, m>. | (g e. Grp /\ E.s(m:(ran g X. s)-->s /\ A.x e. s (((Id` g)mx) = x /\ A.y e. ran gA.z e. ran g((ygz)mx) = (ym(zmx)))))}
3		df-ga 9448	. . . . . . . 8 |- GrpAct = {<.g, m>. | (g e. Grp /\ E.s(m:(ran g X. s)-->s /\ A.x e. s (((Id` g)mx) = x /\ A.y e. ran gA.z e. ran g((ygz)mx) = (ym(zmx)))))}
4	3	releqi 4072	. . . . . . 7 |- (Rel GrpAct <-> Rel {<.g, m>. | (g e. Grp /\ E.s(m:(ran g X. s)-->s /\ A.x e. s (((Id` g)mx) = x /\ A.y e. ran gA.z e. ran g((ygz)mx) = (ym(zmx)))))})
5	2, 4	mpbir 207	. . . . . 6 |- Rel GrpAct
6	5	brrelexi 4029	. . . . 5 |- (GGrpActM -> G e. _V)
7	1, 6	sylbir 218	. . . 4 |- (<.G, M>. e. GrpAct -> G e. _V)
8	7	anim1i 361	. . 3 |- ((<.G, M>. e. GrpAct /\ M e. A) -> (G e. _V /\ M e. A))
9	8	ancoms 484	. 2 |- ((M e. A /\ <.G, M>. e. GrpAct) -> (G e. _V /\ M e. A))
10		elisset 2299	. . . . 5 |- (G e. Grp -> G e. _V)
11	10	adantr 425	. . . 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))))) -> G e. _V)
12	11	anim1i 361	. . 3 |- (((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. _V /\ M e. A))
13	12	ancoms 484	. 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. _V /\ M e. A))
14		eleq1 1957	. . . . 5 |- (g = G -> (g e. Grp <-> G e. Grp))
15		rneq 4186	. . . . . . . . 9 |- (g = G -> ran g = ran G)
16		isgalem.1	. . . . . . . . 9 |- X = ran G
17	15, 16	syl6eqr 1946	. . . . . . . 8 |- (g = G -> ran g = X)
18		xpeq1 4016	. . . . . . . 8 |- (ran g = X -> (ran g X. s) = (X X. s))
19		feq2 4552	. . . . . . . 8 |- ((ran g X. s) = (X X. s) -> (m:(ran g X. s)-->s <-> m:(X X. s)-->s))
20	17, 18, 19	3syl 24	. . . . . . 7 |- (g = G -> (m:(ran g X. s)-->s <-> m:(X X. s)-->s))
21		fveq2 4681	. . . . . . . . . . . 12 |- (g = G -> (Id` g) = (Id` G))
22		isgalem.2	. . . . . . . . . . . 12 |- U = (Id` G)
23	21, 22	syl6eqr 1946	. . . . . . . . . . 11 |- (g = G -> (Id` g) = U)
24	23	opreq1d 4897	. . . . . . . . . 10 |- (g = G -> ((Id` g)mx) = (Umx))
25	24	eqeq1d 1892	. . . . . . . . 9 |- (g = G -> (((Id` g)mx) = x <-> (Umx) = x))
26		opreq 4888	. . . . . . . . . . . . 13 |- (g = G -> (ygz) = (yGz))
27	26	opreq1d 4897	. . . . . . . . . . . 12 |- (g = G -> ((ygz)mx) = ((yGz)mx))
28	27	eqeq1d 1892	. . . . . . . . . . 11 |- (g = G -> (((ygz)mx) = (ym(zmx)) <-> ((yGz)mx) = (ym(zmx))))
29	17, 28	raleqbidv 2274	. . . . . . . . . 10 |- (g = G -> (A.z e. ran g((ygz)mx) = (ym(zmx)) <-> A.z e. X ((yGz)mx) = (ym(zmx))))
30	17, 29	raleqbidv 2274	. . . . . . . . 9 |- (g = G -> (A.y e. ran gA.z e. ran g((ygz)mx) = (ym(zmx)) <-> A.y e. X A.z e. X ((yGz)mx) = (ym(zmx))))
31	25, 30	anbi12d 690	. . . . . . . 8 |- (g = G -> ((((Id` g)mx) = x /\ A.y e. ran gA.z e. ran g((ygz)mx) = (ym(zmx))) <-> ((Umx) = x /\ A.y e. X A.z e. X ((yGz)mx) = (ym(zmx)))))
32	31	ralbidv 2123	. . . . . . 7 |- (g = G -> (A.x e. s (((Id` g)mx) = x /\ A.y e. ran gA.z e. ran g((ygz)mx) = (ym(zmx))) <-> A.x e. s ((Umx) = x /\ A.y e. X A.z e. X ((yGz)mx) = (ym(zmx)))))
33	20, 32	anbi12d 690	. . . . . 6 |- (g = G -> ((m:(ran g X. s)-->s /\ A.x e. s (((Id` g)mx) = x /\ A.y e. ran gA.z e. ran g((ygz)mx) = (ym(zmx)))) <-> (m:(X X. s)-->s /\ A.x e. s ((Umx) = x /\ A.y e. X A.z e. X ((yGz)mx) = (ym(zmx))))))
34	33	exbidv 1657	. . . . 5 |- (g = G -> (E.s(m:(ran g X. s)-->s /\ A.x e. s (((Id` g)mx) = x /\ A.y e. ran gA.z e. ran g((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	14, 34	anbi12d 690	. . . 4 |- (g = G -> ((g e. Grp /\ E.s(m:(ran g X. s)-->s /\ A.x e. s (((Id` g)mx) = x /\ A.y e. ran gA.z e. ran g((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		feq1 4551	. . . . . . 7 |- (m = M -> (m:(X X. s)-->s <-> M:(X X. s)-->s))
37		opreq 4888	. . . . . . . . . 10 |- (m = M -> (Umx) = (UMx))
38	37	eqeq1d 1892	. . . . . . . . 9 |- (m = M -> ((Umx) = x <-> (UMx) = x))
39		opreq 4888	. . . . . . . . . . 11 |- (m = M -> ((yGz)mx) = ((yGz)Mx))
40		opreq 4888	. . . . . . . . . . . 12 |- (m = M -> (ym(zmx)) = (yM(zmx)))
41		opreq 4888	. . . . . . . . . . . . 13 |- (m = M -> (zmx) = (zMx))
42	41	opreq2d 4898	. . . . . . . . . . . 12 |- (m = M -> (yM(zmx)) = (yM(zMx)))
43	40, 42	eqtrd 1925	. . . . . . . . . . 11 |- (m = M -> (ym(zmx)) = (yM(zMx)))
44	39, 43	eqeq12d 1899	. . . . . . . . . 10 |- (m = M -> (((yGz)mx) = (ym(zmx)) <-> ((yGz)Mx) = (yM(zMx))))
45	44	2ralbidv 2140	. . . . . . . . 9 |- (m = M -> (A.y e. X A.z e. X ((yGz)mx) = (ym(zmx)) <-> A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))
46	38, 45	anbi12d 690	. . . . . . . 8 |- (m = M -> (((Umx) = x /\ A.y e. X A.z e. X ((yGz)mx) = (ym(zmx))) <-> ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))
47	46	ralbidv 2123	. . . . . . 7 |- (m = M -> (A.x e. s ((Umx) = x /\ A.y e. X A.z e. X ((yGz)mx) = (ym(zmx))) <-> A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))
48	36, 47	anbi12d 690	. . . . . 6 |- (m = M -> ((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. s)-->s /\ A.x e. s ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
49	48	exbidv 1657	. . . . 5 |- (m = M -> (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)))) <-> 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))))))
50	49	anbi2d 678	. . . 4 |- (m = M -> ((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 /\ 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)))))))
51	35, 50	opelopabg 3567	. . 3 |- ((G e. _V /\ M e. A) -> (<.G, M>. e. {<.g, m>. | (g e. Grp /\ E.s(m:(ran g X. s)-->s /\ A.x e. s (((Id` g)mx) = x /\ A.y e. ran gA.z e. ran g((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)))))))
52	3	eleq2i 1961	. . 3 |- (<.G, M>. e. GrpAct <-> <.G, M>. e. {<.g, m>. | (g e. Grp /\ E.s(m:(ran g X. s)-->s /\ A.x e. s (((Id` g)mx) = x /\ A.y e. ran gA.z e. ran g((ygz)mx) = (ym(zmx)))))})
53	51, 52	syl5bb 591	. 2 |- ((G e. _V /\ 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)))))))
54	9, 13, 53	pm5.21nd 744	1 |- (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)))))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  _Vcvv 2292  <.cop 3046   class class class wbr 3338  {copab 3395   X. cxp 3984  ran crn 3987  Rel wrel 3991  -->wf 3994  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  GrpActcga 9447

This theorem is referenced by:  isga 9450

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-fv 4014  df-opr 4886  df-ga 9448

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