Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ablprop Unicode version

Theorem ablprop 15116
 Description: If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)
Hypotheses
Ref Expression
ablprop.b
ablprop.p
Assertion
Ref Expression
ablprop

Proof of Theorem ablprop
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2297 . . 3
2 ablprop.b . . . 4
32a1i 10 . . 3
4 ablprop.p . . . . 5
54oveqi 5887 . . . 4
65a1i 10 . . 3
71, 3, 6ablpropd 15115 . 2
87trud 1314 1
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358   wtru 1307   wceq 1632   wcel 1696  cfv 5271  (class class class)co 5874  cbs 13164   cplusg 13224  cabel 15106 This theorem is referenced by:  zlmlmod  16493  dvaabl  31836 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
 Copyright terms: Public domain W3C validator