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Theorem ablpropd 15115
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 14516 . . 3  |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
51, 2, 3cmnpropd 15114 . . 3  |-  ( ph  ->  ( K  e. CMnd  <->  L  e. CMnd ) )
64, 5anbi12d 691 . 2  |-  ( ph  ->  ( ( K  e. 
Grp  /\  K  e. CMnd )  <-> 
( L  e.  Grp  /\  L  e. CMnd ) ) )
7 isabl 15109 . 2  |-  ( K  e.  Abel  <->  ( K  e. 
Grp  /\  K  e. CMnd ) )
8 isabl 15109 . 2  |-  ( L  e.  Abel  <->  ( L  e. 
Grp  /\  L  e. CMnd ) )
96, 7, 83bitr4g 279 1  |-  ( ph  ->  ( K  e.  Abel  <->  L  e.  Abel ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378  CMndccmn 15105   Abelcabel 15106
This theorem is referenced by:  ablprop  15116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-0g 13420  df-mnd 14383  df-grp 14505  df-cmn 15107  df-abl 15108
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