Theorem isablNEW 17135

Description: The predicate "is an Abelian (commutative) group."

Hypotheses

Ref	Expression
isabl.1NEW	|- B = (base` G)
isabl.2NEW	|- P = (+g` G)

Assertion

Ref	Expression
isablNEW	|- (G e. AbelNEW <-> (G e. GrpNEW /\ A.x e. B A.y e. B (xPy) = (yPx)))

Distinct variable groups:   x,y,B   x,G,y

Proof of Theorem isablNEW

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . 5 |- (g = G -> (base` g) = (base` G))
2		isabl.1NEW	. . . . 5 |- B = (base` G)
3	1, 2	syl6eqr 1946	. . . 4 |- (g = G -> (base` g) = B)
4		raleq 2266	. . . . 5 |- ((base` g) = B -> (A.y e. (base` g)(x(+g` g)y) = (y(+g` g)x) <-> A.y e. B (x(+g` g)y) = (y(+g` g)x)))
5	4	raleqbi1dv 2271	. . . 4 |- ((base` g) = B -> (A.x e. (base` g)A.y e. (base` g)(x(+g` g)y) = (y(+g` g)x) <-> A.x e. B A.y e. B (x(+g` g)y) = (y(+g` g)x)))
6	3, 5	syl 12	. . 3 |- (g = G -> (A.x e. (base` g)A.y e. (base` g)(x(+g` g)y) = (y(+g` g)x) <-> A.x e. B A.y e. B (x(+g` g)y) = (y(+g` g)x)))
7		fveq2 4681	. . . . . . 7 |- (g = G -> (+g` g) = (+g` G))
8		isabl.2NEW	. . . . . . 7 |- P = (+g` G)
9	7, 8	syl6eqr 1946	. . . . . 6 |- (g = G -> (+g` g) = P)
10	9	opreqd 4899	. . . . 5 |- (g = G -> (x(+g` g)y) = (xPy))
11	9	opreqd 4899	. . . . 5 |- (g = G -> (y(+g` g)x) = (yPx))
12	10, 11	eqeq12d 1899	. . . 4 |- (g = G -> ((x(+g` g)y) = (y(+g` g)x) <-> (xPy) = (yPx)))
13	12	2ralbidv 2140	. . 3 |- (g = G -> (A.x e. B A.y e. B (x(+g` g)y) = (y(+g` g)x) <-> A.x e. B A.y e. B (xPy) = (yPx)))
14	6, 13	bitrd 587	. 2 |- (g = G -> (A.x e. (base` g)A.y e. (base` g)(x(+g` g)y) = (y(+g` g)x) <-> A.x e. B A.y e. B (xPy) = (yPx)))
15		df-ablNEW 17092	. 2 |- AbelNEW = {g e. GrpNEW | A.x e. (base` g)A.y e. (base` g)(x(+g` g)y) = (y(+g` g)x)}
16	14, 15	elrab2 2416	1 |- (G e. AbelNEW <-> (G e. GrpNEW /\ A.x e. B A.y e. B (xPy) = (yPx)))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  ` cfv 3998  (class class class)co 4884  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081  AbelNEWcabel 17084

This theorem is referenced by:  isabliNEW 17136  ablgrpNEW 17137  ablcomNEW 17138

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-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-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-ablNEW 17092

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