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Theorem grpomndo 25220
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 2443 . . . . 5  |-  ran  G  =  ran  G
21isgrpo 25070 . . . 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 25082 . . . . . . . 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 2836 . . . . . . . . . 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 2911 . . . . . . . . 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 25219 . . . . . . . . . . . . 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 1196 . . . . . . . . . . 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 1191 . . 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 974    = wceq 1383    e. wcel 1804   A.wral 2793   E.wrex 2794    X. cxp 4987   ran crn 4990   -->wf 5574  (class class class)co 6281   GrpOpcgr 25060  MndOpcmndo 25211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-ov 6284  df-grpo 25065  df-ass 25187  df-exid 25189  df-mgmOLD 25193  df-sgrOLD 25205  df-mndo 25212
This theorem is referenced by:  isdrngo2  30336
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