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Theorem mndoass2 24772
Description: A monoid is associative. (Contributed by FL, 12-Dec-2009.)
Hypothesis
Ref Expression
mndoid.1  |-  X  =  ran  G
Assertion
Ref Expression
mndoass2  |-  ( ( G  e. MndOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )

Proof of Theorem mndoass2
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndoid.1 . . 3  |-  X  =  ran  G
21mndoass 24771 . 2  |-  ( G  e. MndOp  ->  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )
3 oveq1 5865 . . . . 5  |-  ( x  =  A  ->  (
x G y )  =  ( A G y ) )
43oveq1d 5873 . . . 4  |-  ( x  =  A  ->  (
( x G y ) G z )  =  ( ( A G y ) G z ) )
5 oveq1 5865 . . . 4  |-  ( x  =  A  ->  (
x G ( y G z ) )  =  ( A G ( y G z ) ) )
64, 5eqeq12d 2297 . . 3  |-  ( x  =  A  ->  (
( ( x G y ) G z )  =  ( x G ( y G z ) )  <->  ( ( A G y ) G z )  =  ( A G ( y G z ) ) ) )
7 oveq2 5866 . . . . 5  |-  ( y  =  B  ->  ( A G y )  =  ( A G B ) )
87oveq1d 5873 . . . 4  |-  ( y  =  B  ->  (
( A G y ) G z )  =  ( ( A G B ) G z ) )
9 oveq1 5865 . . . . 5  |-  ( y  =  B  ->  (
y G z )  =  ( B G z ) )
109oveq2d 5874 . . . 4  |-  ( y  =  B  ->  ( A G ( y G z ) )  =  ( A G ( B G z ) ) )
118, 10eqeq12d 2297 . . 3  |-  ( y  =  B  ->  (
( ( A G y ) G z )  =  ( A G ( y G z ) )  <->  ( ( A G B ) G z )  =  ( A G ( B G z ) ) ) )
12 oveq2 5866 . . . 4  |-  ( z  =  C  ->  (
( A G B ) G z )  =  ( ( A G B ) G C ) )
13 oveq2 5866 . . . . 5  |-  ( z  =  C  ->  ( B G z )  =  ( B G C ) )
1413oveq2d 5874 . . . 4  |-  ( z  =  C  ->  ( A G ( B G z ) )  =  ( A G ( B G C ) ) )
1512, 14eqeq12d 2297 . . 3  |-  ( z  =  C  ->  (
( ( A G B ) G z )  =  ( A G ( B G z ) )  <->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) ) )
166, 11, 15rspc3v 2893 . 2  |-  ( ( A  e.  X  /\  B  e.  X  /\  C  e.  X )  ->  ( A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) ) )
172, 16mpan9 455 1  |-  ( ( G  e. MndOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ran crn 4690  (class class class)co 5858  MndOpcmndo 21004
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-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005
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