Theorem isabld 18206

Description: Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)

Hypotheses

Ref	Expression
isabld.b	|- ( ph -> B = ( Base ` G ) )
isabld.p	|- ( ph -> .+ = ( +g ` G ) )
isabld.g	|- ( ph -> G e. Grp )
isabld.c	|- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )

Assertion

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

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

Allowed substitution hints:   .+ ( x, y)

Proof of Theorem isabld

Step	Hyp	Ref	Expression
1		isabld.g	. 2 |- ( ph -> G e. Grp )
2		isabld.b	. . 3 |- ( ph -> B = ( Base ` G ) )
3		isabld.p	. . 3 |- ( ph -> .+ = ( +g ` G ) )
4		grpmnd 17429	. . . 4 |- ( G e. Grp -> G e. Mnd )
5	1, 4	syl 17	. . 3 |- ( ph -> G e. Mnd )
6		isabld.c	. . 3 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .+ y ) = ( y .+ x ) )
7	2, 3, 5, 6	iscmnd 18205	. 2 |- ( ph -> G e. CMnd )
8		isabl 18197	. 2 |- ( G e. Abel <-> ( G e. Grp /\ G e. CMnd ) )
9	1, 7, 8	sylanbrc 698	1 |- ( ph -> G e. Abel )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294   Grpcgrp 17422  CMndccmn 18193   Abelcabl 18194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-grp 17425  df-cmn 18195  df-abl 18196

This theorem is referenced by:  subgabl  18241  gex2abl  18254  cygabl  18292  ringabl  18580  lmodabl  18910  dchrabl  24979  tgrpabl  36039  erngdvlem2N  36277  erngdvlem2-rN  36285