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

Theorem isablod 20967
Description: Properties that determine an Abelian group operation. (Changed label from isabld 15102 to isablod 20967-NM 6-Aug-2013.) (Contributed by Jeff Madsen, 5-Dec-2009.) (New usage is discouraged.)
Hypotheses
Ref Expression
isgrpda.1  |-  ( ph  ->  X  e.  _V )
isgrpda.2  |-  ( ph  ->  G : ( X  X.  X ) --> X )
isgrpda.3  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  -> 
( ( x G y ) G z )  =  ( x G ( y G z ) ) )
isgrpda.4  |-  ( ph  ->  U  e.  X )
isgrpda.5  |-  ( (
ph  /\  x  e.  X )  ->  ( U G x )  =  x )
isabld.6  |-  ( (
ph  /\  x  e.  X )  ->  N  e.  X )
isabld.7  |-  ( (
ph  /\  x  e.  X )  ->  ( N G x )  =  U )
isabld.8  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x G y )  =  ( y G x ) )
Assertion
Ref Expression
isablod  |-  ( ph  ->  G  e.  AbelOp )
Distinct variable groups:    ph, x, y, z    x, G, y, z    x, X, y, z    x, U, y, z
Allowed substitution hints:    N( x, y, z)

Proof of Theorem isablod
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 isgrpda.1 . 2  |-  ( ph  ->  X  e.  _V )
2 isgrpda.2 . 2  |-  ( ph  ->  G : ( X  X.  X ) --> X )
3 isgrpda.3 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  -> 
( ( x G y ) G z )  =  ( x G ( y G z ) ) )
4 isgrpda.4 . 2  |-  ( ph  ->  U  e.  X )
5 isgrpda.5 . 2  |-  ( (
ph  /\  x  e.  X )  ->  ( U G x )  =  x )
6 isabld.6 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  N  e.  X )
7 isabld.7 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( N G x )  =  U )
8 oveq1 5865 . . . . 5  |-  ( n  =  N  ->  (
n G x )  =  ( N G x ) )
98eqeq1d 2291 . . . 4  |-  ( n  =  N  ->  (
( n G x )  =  U  <->  ( N G x )  =  U ) )
109rspcev 2884 . . 3  |-  ( ( N  e.  X  /\  ( N G x )  =  U )  ->  E. n  e.  X  ( n G x )  =  U )
116, 7, 10syl2anc 642 . 2  |-  ( (
ph  /\  x  e.  X )  ->  E. n  e.  X  ( n G x )  =  U )
12 isabld.8 . 2  |-  ( (
ph  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x G y )  =  ( y G x ) )
131, 2, 3, 4, 5, 11, 12isabloda 20966 1  |-  ( ph  ->  G  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    X. cxp 4687   -->wf 5251  (class class class)co 5858   AbelOpcablo 20948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-grpo 20858  df-ablo 20949
  Copyright terms: Public domain W3C validator