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Theorem grpomndo 25012
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 2462 . . . . 5  |-  ran  G  =  ran  G
21isgrpo 24862 . . . 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 207 . . 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 24874 . . . . . . . 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 457 . . . . . . . . . . 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 2852 . . . . . . . . . 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 2927 . . . . . . . . 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 25011 . . . . . . . . . . . . 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 225 . . . . . . . . . . . 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 1190 . . . . . . . . . . 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 430 . . . . . . . . . 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 81 . . . . . . . . 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 16 . . . . . . . 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 36 . . . . . . 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 ) )
1514expdcom 439 . . . . . 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 11 . . . . 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 80 . . . 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 1185 . . 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 37 . 2  |-  ( G  e.  GrpOp  ->  ( G  e.  GrpOp  ->  G  e. MndOp ) )
2019pm2.43i 47 1  |-  ( G  e.  GrpOp  ->  G  e. MndOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810    X. cxp 4992   ran crn 4995   -->wf 5577  (class class class)co 6277   GrpOpcgr 24852  MndOpcmndo 25003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-ov 6280  df-grpo 24857  df-ass 24979  df-exid 24981  df-mgm 24985  df-sgr 24997  df-mndo 25004
This theorem is referenced by:  isdrngo2  29953
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