Theorem ablcomgrp 14702

Description: An abelian group is a commutative group.

Assertion

Ref	Expression
ablcomgrp	|- (G e. Abel -> G e. (Grp i^i Com1))

Proof of Theorem ablcomgrp

Step	Hyp	Ref	Expression
1		grprndm 9334	. . . 4 |- (G e. Grp -> ran G = dom dom G)
2		eleq2 1958	. . . . . . . 8 |- (ran G = dom dom G -> (x e. ran G <-> x e. dom dom G))
3		raleq 2266	. . . . . . . 8 |- (ran G = dom dom G -> (A.y e. ran G(xGy) = (yGx) <-> A.y e. dom dom G(xGy) = (yGx)))
4	2, 3	imbi12d 688	. . . . . . 7 |- (ran G = dom dom G -> ((x e. ran G -> A.y e. ran G(xGy) = (yGx)) <-> (x e. dom dom G -> A.y e. dom dom G(xGy) = (yGx))))
5	4	ralbidv2 2125	. . . . . 6 |- (ran G = dom dom G -> (A.x e. ran GA.y e. ran G(xGy) = (yGx) <-> A.x e. dom dom GA.y e. dom dom G(xGy) = (yGx)))
6	5	imbi1d 675	. . . . 5 |- (ran G = dom dom G -> ((A.x e. ran GA.y e. ran G(xGy) = (yGx) -> G e. Com1) <-> (A.x e. dom dom GA.y e. dom dom G(xGy) = (yGx) -> G e. Com1)))
7		eqid 1884	. . . . . . 7 |- dom dom G = dom dom G
8	7	iscom 14689	. . . . . 6 |- (G e. Grp -> (G e. Com1 <-> A.x e. dom dom GA.y e. dom dom G(xGy) = (yGx)))
9	8	biimprd 171	. . . . 5 |- (G e. Grp -> (A.x e. dom dom GA.y e. dom dom G(xGy) = (yGx) -> G e. Com1))
10	6, 9	syl5bir 227	. . . 4 |- (ran G = dom dom G -> (G e. Grp -> (A.x e. ran GA.y e. ran G(xGy) = (yGx) -> G e. Com1)))
11	1, 10	mpcom 60	. . 3 |- (G e. Grp -> (A.x e. ran GA.y e. ran G(xGy) = (yGx) -> G e. Com1))
12	11	imdistani 491	. 2 |- ((G e. Grp /\ A.x e. ran GA.y e. ran G(xGy) = (yGx)) -> (G e. Grp /\ G e. Com1))
13		eqid 1884	. . 3 |- ran G = ran G
14	13	isabl 9409	. 2 |- (G e. Abel <-> (G e. Grp /\ A.x e. ran GA.y e. ran G(xGy) = (yGx)))
15		elin 2786	. 2 |- (G e. (Grp i^i Com1) <-> (G e. Grp /\ G e. Com1))
16	12, 14, 15	3imtr4i 236	1 |- (G e. Abel -> G e. (Grp i^i Com1))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   i^i cin 2592  dom cdm 3986  ran crn 3987  (class class class)co 4884  Grpcgr 9311  Abelcabl 9407  Com1ccm1 14687

This theorem is referenced by:  fprodsub 14742

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-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  df-com1 14688

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