Theorem isablda 16035

Description: Properties that determine an Abelian group operation.

Hypotheses

Ref	Expression
isgrpda.1	|- (ph -> X e. _V)
isgrpda.2	|- (ph -> G:(X X. X)-->X)
isgrpda.3	|- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy)Gz) = (xG(yGz)))
isgrpda.4	|- (ph -> U e. X)
isgrpda.5	|- ((ph /\ x e. X) -> (UGx) = x)
isablda.6	|- ((ph /\ x e. X) -> E.n e. X (nGx) = U)
isablda.7	|- ((ph /\ (x e. X /\ y e. X)) -> (xGy) = (yGx))

Assertion

Ref	Expression
isablda	|- (ph -> G e. Abel)

Distinct variable groups:   ph,x,y,z   n,G,x,y,z   n,X,x,y,z   U,n,x,y,z

Proof of Theorem isablda

Step	Hyp	Ref	Expression
1		eqid 1884	. . 3 |- ran G = ran G
2	1	isabl 9409	. 2 |- (G e. Abel <-> (G e. Grp /\ A.x e. ran GA.y e. ran G(xGy) = (yGx)))
3		isgrpda.1	. . 3 |- (ph -> X e. _V)
4		isgrpda.2	. . 3 |- (ph -> G:(X X. X)-->X)
5		isgrpda.3	. . 3 |- ((ph /\ (x e. X /\ y e. X /\ z e. X)) -> ((xGy)Gz) = (xG(yGz)))
6		isgrpda.4	. . 3 |- (ph -> U e. X)
7		isgrpda.5	. . 3 |- ((ph /\ x e. X) -> (UGx) = x)
8		isablda.6	. . 3 |- ((ph /\ x e. X) -> E.n e. X (nGx) = U)
9	3, 4, 5, 6, 7, 8	isgrpda 16033	. 2 |- (ph -> G e. Grp)
10		grprndm 9334	. . . . . . . 8 |- (G e. Grp -> ran G = dom dom G)
11	9, 10	syl 12	. . . . . . 7 |- (ph -> ran G = dom dom G)
12		fdm 4567	. . . . . . . . . 10 |- (G:(X X. X)-->X -> dom G = (X X. X))
13	4, 12	syl 12	. . . . . . . . 9 |- (ph -> dom G = (X X. X))
14	13	dmeqd 4159	. . . . . . . 8 |- (ph -> dom dom G = dom ( X X. X))
15		dmxpid 4179	. . . . . . . 8 |- dom ( X X. X) = X
16	14, 15	syl6eq 1944	. . . . . . 7 |- (ph -> dom dom G = X)
17	11, 16	eqtrd 1925	. . . . . 6 |- (ph -> ran G = X)
18	17	eleq2d 1964	. . . . 5 |- (ph -> (x e. ran G <-> x e. X))
19	17	eleq2d 1964	. . . . 5 |- (ph -> (y e. ran G <-> y e. X))
20	18, 19	anbi12d 690	. . . 4 |- (ph -> ((x e. ran G /\ y e. ran G) <-> (x e. X /\ y e. X)))
21		isablda.7	. . . . 5 |- ((ph /\ (x e. X /\ y e. X)) -> (xGy) = (yGx))
22	21	ex 402	. . . 4 |- (ph -> ((x e. X /\ y e. X) -> (xGy) = (yGx)))
23	20, 22	sylbid 220	. . 3 |- (ph -> ((x e. ran G /\ y e. ran G) -> (xGy) = (yGx)))
24	23	r19.21aivv 2183	. 2 |- (ph -> A.x e. ran GA.y e. ran G(xGy) = (yGx))
25	2, 9, 24	sylanbrc 527	1 |- (ph -> G e. Abel)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   X. cxp 3984  dom cdm 3986  ran crn 3987  -->wf 3994  (class class class)co 4884  Grpcgr 9311  Abelcabl 9407

This theorem is referenced by:  isabld 16036

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-rab 2112  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-abl 9408

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