Theorem ltlga 14729

Description: A group operation is a left group action of the group on itself.

Assertion

Ref	Expression
ltlga	|- (G e. Grp -> <.G, G>. e. GrpAct)

Proof of Theorem ltlga

Step	Hyp	Ref	Expression
1		id 73	. . 3 |- (G e. Grp -> G e. Grp)
2		grpmnd 10393	. . . . 5 |- (G e. Grp -> G e. Mnd)
3		eqid 1884	. . . . . 6 |- ran G = ran G
4	3	mndio 14719	. . . . 5 |- (G e. Mnd -> G:(ran G X. ran G)-->ran G)
5		grprndm 9334	. . . . . . . . 9 |- (G e. Grp -> ran G = dom dom G)
6		mndismgm 10388	. . . . . . . . . . 11 |- (G e. Mnd -> G e. Magma)
7	2, 6	syl 12	. . . . . . . . . 10 |- (G e. Grp -> G e. Magma)
8		mgmrddd 14727	. . . . . . . . . 10 |- (G e. Magma -> ran dom G = dom dom G)
9	7, 8	syl 12	. . . . . . . . 9 |- (G e. Grp -> ran dom G = dom dom G)
10	5, 9	eqtr4d 1928	. . . . . . . 8 |- (G e. Grp -> ran G = ran dom G)
11		xpeq2 4017	. . . . . . . 8 |- (ran G = ran dom G -> (ran G X. ran G) = (ran G X. ran dom G))
12	10, 11	syl 12	. . . . . . 7 |- (G e. Grp -> (ran G X. ran G) = (ran G X. ran dom G))
13		feq23 4554	. . . . . . 7 |- (((ran G X. ran G) = (ran G X. ran dom G) /\ ran G = ran dom G) -> (G:(ran G X. ran G)-->ran G <-> G:(ran G X. ran dom G)-->ran dom G))
14	12, 10, 13	syl11anc 524	. . . . . 6 |- (G e. Grp -> (G:(ran G X. ran G)-->ran G <-> G:(ran G X. ran dom G)-->ran dom G))
15	14	biimpcd 172	. . . . 5 |- (G:(ran G X. ran G)-->ran G -> (G e. Grp -> G:(ran G X. ran dom G)-->ran dom G))
16	2, 4, 15	3syl 24	. . . 4 |- (G e. Grp -> (G e. Grp -> G:(ran G X. ran dom G)-->ran dom G))
17	16	pm2.43i 78	. . 3 |- (G e. Grp -> G:(ran G X. ran dom G)-->ran dom G)
18	9, 5	eqtr4d 1928	. . . . . 6 |- (G e. Grp -> ran dom G = ran G)
19	18	eleq2d 1964	. . . . 5 |- (G e. Grp -> (x e. ran dom G <-> x e. ran G))
20		eqid 1884	. . . . . . . 8 |- (Id` G) = (Id` G)
21	3, 20	grplid 9345	. . . . . . 7 |- ((G e. Grp /\ x e. ran G) -> ((Id` G)Gx) = x)
22	3	grpass 9327	. . . . . . . . . . . 12 |- ((G e. Grp /\ (y e. ran G /\ z e. ran G /\ x e. ran G)) -> ((yGz)Gx) = (yG(zGx)))
23	22	3exp2 1086	. . . . . . . . . . 11 |- (G e. Grp -> (y e. ran G -> (z e. ran G -> (x e. ran G -> ((yGz)Gx) = (yG(zGx))))))
24	23	com23 36	. . . . . . . . . 10 |- (G e. Grp -> (z e. ran G -> (y e. ran G -> (x e. ran G -> ((yGz)Gx) = (yG(zGx))))))
25	24	com24 41	. . . . . . . . 9 |- (G e. Grp -> (x e. ran G -> (y e. ran G -> (z e. ran G -> ((yGz)Gx) = (yG(zGx))))))
26	25	imp4b 392	. . . . . . . 8 |- ((G e. Grp /\ x e. ran G) -> ((y e. ran G /\ z e. ran G) -> ((yGz)Gx) = (yG(zGx))))
27	26	r19.21aivv 2183	. . . . . . 7 |- ((G e. Grp /\ x e. ran G) -> A.y e. ran GA.z e. ran G((yGz)Gx) = (yG(zGx)))
28	21, 27	jca 310	. . . . . 6 |- ((G e. Grp /\ x e. ran G) -> (((Id` G)Gx) = x /\ A.y e. ran GA.z e. ran G((yGz)Gx) = (yG(zGx))))
29	28	ex 402	. . . . 5 |- (G e. Grp -> (x e. ran G -> (((Id` G)Gx) = x /\ A.y e. ran GA.z e. ran G((yGz)Gx) = (yG(zGx)))))
30	19, 29	sylbid 220	. . . 4 |- (G e. Grp -> (x e. ran dom G -> (((Id` G)Gx) = x /\ A.y e. ran GA.z e. ran G((yGz)Gx) = (yG(zGx)))))
31	30	r19.21aiv 2175	. . 3 |- (G e. Grp -> A.x e. ran dom G(((Id` G)Gx) = x /\ A.y e. ran GA.z e. ran G((yGz)Gx) = (yG(zGx))))
32	1, 17, 31	3jca 1050	. 2 |- (G e. Grp -> (G e. Grp /\ G:(ran G X. ran dom G)-->ran dom G /\ A.x e. ran dom G(((Id` G)Gx) = x /\ A.y e. ran GA.z e. ran G((yGz)Gx) = (yG(zGx)))))
33		eqid 1884	. . 3 |- ran dom G = ran dom G
34	3, 33, 20	isga 9450	. 2 |- (G e. Grp -> (<.G, G>. e. GrpAct <-> (G e. Grp /\ G:(ran G X. ran dom G)-->ran dom G /\ A.x e. ran dom G(((Id` G)Gx) = x /\ A.y e. ran GA.z e. ran G((yGz)Gx) = (yG(zGx))))))
35	32, 34	mpbird 213	1 |- (G e. Grp -> <.G, G>. e. GrpAct)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  <.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  Magmacmagm 10365  Mndcmnd 10384

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-rep 3428  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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-grp 9316  df-gid 9317  df-ga 9448  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385

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