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

Theorem cmnpropd 15423
 Description: If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ablpropd.1
ablpropd.2
ablpropd.3
Assertion
Ref Expression
cmnpropd CMnd CMnd
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem cmnpropd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablpropd.1 . . . 4
2 ablpropd.2 . . . 4
3 ablpropd.3 . . . 4
41, 2, 3mndpropd 14723 . . 3
53proplem 13917 . . . . . 6
63proplem 13917 . . . . . . 7
76ancom2s 779 . . . . . 6
85, 7eqeq12d 2452 . . . . 5
982ralbidva 2747 . . . 4
101raleqdv 2912 . . . . 5
111, 10raleqbidv 2918 . . . 4
122raleqdv 2912 . . . . 5
132, 12raleqbidv 2918 . . . 4
149, 11, 133bitr3d 276 . . 3
154, 14anbi12d 693 . 2
16 eqid 2438 . . 3
17 eqid 2438 . . 3
1816, 17iscmn 15421 . 2 CMnd
19 eqid 2438 . . 3
20 eqid 2438 . . 3
2119, 20iscmn 15421 . 2 CMnd
2215, 18, 213bitr4g 281 1 CMnd CMnd
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  cfv 5456  (class class class)co 6083  cbs 13471   cplusg 13531  cmnd 14686  CMndccmn 15414 This theorem is referenced by:  ablpropd  15424  crngpropd  15698  prdscrngd  15721 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-nul 4340  ax-pow 4379 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-iota 5420  df-fv 5464  df-ov 6086  df-mnd 14692  df-cmn 15416
 Copyright terms: Public domain W3C validator