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Theorem ablpropd 15424
Description: If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)
Hypotheses
Ref Expression
ablpropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
ablpropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
ablpropd.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
Assertion
Ref Expression
ablpropd  |-  ( ph  ->  ( K  e.  Abel  <->  L  e.  Abel ) )
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y

Proof of Theorem ablpropd
StepHypRef Expression
1 ablpropd.1 . . . 4  |-  ( ph  ->  B  =  ( Base `  K ) )
2 ablpropd.2 . . . 4  |-  ( ph  ->  B  =  ( Base `  L ) )
3 ablpropd.3 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
41, 2, 3grppropd 14825 . . 3  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
51, 2, 3cmnpropd 15423 . . 3  |-  ( ph  ->  ( K  e. CMnd  <->  L  e. CMnd ) )
64, 5anbi12d 693 . 2  |-  ( ph  ->  ( ( K  e. 
Grp  /\  K  e. CMnd )  <-> 
( L  e.  Grp  /\  L  e. CMnd ) ) )
7 isabl 15418 . 2  |-  ( K  e.  Abel  <->  ( K  e. 
Grp  /\  K  e. CMnd ) )
8 isabl 15418 . 2  |-  ( L  e.  Abel  <->  ( L  e. 
Grp  /\  L  e. CMnd ) )
96, 7, 83bitr4g 281 1  |-  ( ph  ->  ( K  e.  Abel  <->  L  e.  Abel ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   Grpcgrp 14687  CMndccmn 15414   Abelcabel 15415
This theorem is referenced by:  ablprop  15425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-0g 13729  df-mnd 14692  df-grp 14814  df-cmn 15416  df-abl 15417
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