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Theorem isgrpod 20981
Description: Properties that determine a group operation. (Renamed from isgrpd 14523 to isgrpod 20981 to prevent naming conflict. -NM 5-Jun-2013) (Contributed by Jeff Madsen, 1-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 )
isgrpod.6  |-  ( (
ph  /\  x  e.  X )  ->  N  e.  X )
isgrpod.7  |-  ( (
ph  /\  x  e.  X )  ->  ( N G x )  =  U )
Assertion
Ref Expression
isgrpod  |-  ( ph  ->  G  e.  GrpOp )
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 isgrpod
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 isgrpod.6 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  N  e.  X )
7 isgrpod.7 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( N G x )  =  U )
8 oveq1 5881 . . . . 5  |-  ( n  =  N  ->  (
n G x )  =  ( N G x ) )
98eqeq1d 2304 . . . 4  |-  ( n  =  N  ->  (
( n G x )  =  U  <->  ( N G x )  =  U ) )
109rspcev 2897 . . 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 )
121, 2, 3, 4, 5, 11isgrpda 20980 1  |-  ( ph  ->  G  e.  GrpOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    X. cxp 4703   -->wf 5267  (class class class)co 5874   GrpOpcgr 20869
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  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-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-grpo 20874
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