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Theorem grpomndo 21013
Description: A group is a monoid. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
grpomndo  |-  ( G  e.  GrpOp  ->  G  e. MndOp )

Proof of Theorem grpomndo
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . 5  |-  ran  G  =  ran  G
21isgrpo 20863 . . . 4  |-  ( G  e.  GrpOp  ->  ( G  e.  GrpOp 
<->  ( G : ( ran  G  X.  ran  G ) --> ran  G  /\  A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  /\  E. w  e.  ran  G A. x  e.  ran  G ( ( w G x )  =  x  /\  E. y  e. 
ran  G ( y G x )  =  w ) ) ) )
32biimpd 198 . . 3  |-  ( G  e.  GrpOp  ->  ( G  e.  GrpOp  ->  ( G : ( ran  G  X.  ran  G ) --> ran 
G  /\  A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  /\  E. w  e.  ran  G A. x  e.  ran  G ( ( w G x )  =  x  /\  E. y  e.  ran  G ( y G x )  =  w ) ) ) )
41grpoidinv 20875 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  E. x  e.  ran  G A. y  e.  ran  G ( ( ( x G y )  =  y  /\  ( y G x )  =  y )  /\  E. w  e. 
ran  G ( ( w G y )  =  x  /\  (
y G w )  =  x ) ) )
5 simpl 443 . . . . . . . . . . 11  |-  ( ( ( ( x G y )  =  y  /\  ( y G x )  =  y )  /\  E. w  e.  ran  G ( ( w G y )  =  x  /\  (
y G w )  =  x ) )  ->  ( ( x G y )  =  y  /\  ( y G x )  =  y ) )
65ralimi 2618 . . . . . . . . . 10  |-  ( A. y  e.  ran  G ( ( ( x G y )  =  y  /\  ( y G x )  =  y )  /\  E. w  e.  ran  G ( ( w G y )  =  x  /\  (
y G w )  =  x ) )  ->  A. y  e.  ran  G ( ( x G y )  =  y  /\  ( y G x )  =  y ) )
76reximi 2650 . . . . . . . . 9  |-  ( E. x  e.  ran  G A. y  e.  ran  G ( ( ( x G y )  =  y  /\  ( y G x )  =  y )  /\  E. w  e.  ran  G ( ( w G y )  =  x  /\  ( y G w )  =  x ) )  ->  E. x  e.  ran  G A. y  e.  ran  G ( ( x G y )  =  y  /\  (
y G x )  =  y ) )
81ismndo2 21012 . . . . . . . . . . . . 13  |-  ( G  e.  GrpOp  ->  ( G  e. MndOp  <-> 
( G : ( ran  G  X.  ran  G ) --> ran  G  /\  A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  /\  E. x  e.  ran  G A. y  e.  ran  G ( ( x G y )  =  y  /\  ( y G x )  =  y ) ) ) )
98biimprcd 216 . . . . . . . . . . . 12  |-  ( ( G : ( ran 
G  X.  ran  G
) --> ran  G  /\  A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  /\  E. x  e.  ran  G A. y  e.  ran  G ( ( x G y )  =  y  /\  ( y G x )  =  y ) )  ->  ( G  e.  GrpOp  ->  G  e. MndOp ) )
1093exp 1150 . . . . . . . . . . 11  |-  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  ( A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  ->  ( E. x  e.  ran  G A. y  e.  ran  G ( ( x G y )  =  y  /\  ( y G x )  =  y )  ->  ( G  e.  GrpOp  ->  G  e. MndOp ) ) ) )
1110impcom 419 . . . . . . . . . 10  |-  ( ( A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  /\  G : ( ran  G  X.  ran  G ) --> ran 
G )  ->  ( E. x  e.  ran  G A. y  e.  ran  G ( ( x G y )  =  y  /\  ( y G x )  =  y )  ->  ( G  e.  GrpOp  ->  G  e. MndOp ) ) )
1211com3l 75 . . . . . . . . 9  |-  ( E. x  e.  ran  G A. y  e.  ran  G ( ( x G y )  =  y  /\  ( y G x )  =  y )  ->  ( G  e.  GrpOp  ->  ( ( A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  /\  G : ( ran  G  X.  ran  G ) --> ran 
G )  ->  G  e. MndOp ) ) )
137, 12syl 15 . . . . . . . 8  |-  ( E. x  e.  ran  G A. y  e.  ran  G ( ( ( x G y )  =  y  /\  ( y G x )  =  y )  /\  E. w  e.  ran  G ( ( w G y )  =  x  /\  ( y G w )  =  x ) )  ->  ( G  e.  GrpOp  ->  ( ( A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  /\  G : ( ran  G  X.  ran  G ) --> ran 
G )  ->  G  e. MndOp ) ) )
144, 13mpcom 32 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( ( A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  /\  G : ( ran  G  X.  ran  G ) --> ran 
G )  ->  G  e. MndOp ) )
1514exp3acom3r 1360 . . . . . 6  |-  ( A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  ->  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  ( G  e.  GrpOp  ->  G  e. MndOp ) ) )
1615a1i 10 . . . . 5  |-  ( E. w  e.  ran  G A. x  e.  ran  G ( ( w G x )  =  x  /\  E. y  e. 
ran  G ( y G x )  =  w )  ->  ( A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  -> 
( G : ( ran  G  X.  ran  G ) --> ran  G  ->  ( G  e.  GrpOp  ->  G  e. MndOp ) ) ) )
1716com13 74 . . . 4  |-  ( G : ( ran  G  X.  ran  G ) --> ran 
G  ->  ( A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  ->  ( E. w  e.  ran  G A. x  e.  ran  G ( ( w G x )  =  x  /\  E. y  e. 
ran  G ( y G x )  =  w )  ->  ( G  e.  GrpOp  ->  G  e. MndOp ) ) ) )
18173imp 1145 . . 3  |-  ( ( G : ( ran 
G  X.  ran  G
) --> ran  G  /\  A. x  e.  ran  G A. y  e.  ran  G A. z  e.  ran  G ( ( x G y ) G z )  =  ( x G ( y G z ) )  /\  E. w  e.  ran  G A. x  e.  ran  G ( ( w G x )  =  x  /\  E. y  e. 
ran  G ( y G x )  =  w ) )  -> 
( G  e.  GrpOp  ->  G  e. MndOp ) )
193, 18syli 33 . 2  |-  ( G  e.  GrpOp  ->  ( G  e.  GrpOp  ->  G  e. MndOp ) )
2019pm2.43i 43 1  |-  ( G  e.  GrpOp  ->  G  e. MndOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    X. cxp 4687   ran crn 4690   -->wf 5251  (class class class)co 5858   GrpOpcgr 20853  MndOpcmndo 21004
This theorem is referenced by:  clfsebs4  25381  zintdom  25438  svli2  25484  isdrngo2  26589
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-sep 4141  ax-nul 4149  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-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-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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-grpo 20858  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005
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