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Theorem grpomndo 21926
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 2435 . . . . 5  |-  ran  G  =  ran  G
21isgrpo 21776 . . . 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 199 . . 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 21788 . . . . . . . 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 444 . . . . . . . . . . 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 2773 . . . . . . . . . 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 2805 . . . . . . . . 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 21925 . . . . . . . . . . . . 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 217 . . . . . . . . . . . 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 1152 . . . . . . . . . . 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 420 . . . . . . . . . 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 77 . . . . . . . . 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 34 . . . . . . 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 1379 . . . . . 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 76 . . . 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 1147 . . 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 35 . 2  |-  ( G  e.  GrpOp  ->  ( G  e.  GrpOp  ->  G  e. MndOp ) )
2019pm2.43i 45 1  |-  ( G  e.  GrpOp  ->  G  e. MndOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698    X. cxp 4868   ran crn 4871   -->wf 5442  (class class class)co 6073   GrpOpcgr 21766  MndOpcmndo 21917
This theorem is referenced by:  isdrngo2  26565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-grpo 21771  df-ass 21893  df-exid 21895  df-mgm 21899  df-sgr 21911  df-mndo 21918
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