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Theorem grpomndo 26016
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 2428 . . . . 5  |-  ran  G  =  ran  G
21isgrpo 25866 . . . 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 210 . . 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 25878 . . . . . . . 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 458 . . . . . . . . . . 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 2758 . . . . . . . . . 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 2832 . . . . . . . . 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 26015 . . . . . . . . . . . . 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 228 . . . . . . . . . . . 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 1204 . . . . . . . . . . 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 431 . . . . . . . . . 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 84 . . . . . . . . 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 17 . . . . . . . 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 37 . . . . . . 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 440 . . . . . 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 83 . . . 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 1199 . . 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 38 . 2  |-  ( G  e.  GrpOp  ->  ( G  e.  GrpOp  ->  G  e. MndOp ) )
2019pm2.43i 49 1  |-  ( G  e.  GrpOp  ->  G  e. MndOp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   A.wral 2714   E.wrex 2715    X. cxp 4794   ran crn 4797   -->wf 5540  (class class class)co 6249   GrpOpcgr 25856  MndOpcmndo 26007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-fo 5550  df-fv 5552  df-ov 6252  df-grpo 25861  df-ass 25983  df-exid 25985  df-mgmOLD 25989  df-sgrOLD 26001  df-mndo 26008
This theorem is referenced by:  isdrngo2  32104
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