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Theorem cmpasso 25185
Description: The 9th "axiom" of a category:  ( o_ `  T ) is associative. (Contributed by FL, 29-Oct-2007.)
Hypotheses
Ref Expression
cmpasso.1  |-  M  =  dom  D
cmpasso.2  |-  D  =  ( dom_ `  T
)
cmpasso.5  |-  C  =  ( cod_ `  T
)
cmpasso.6  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
cmpasso  |-  ( ( T  e.  Cat OLD  /\  ( F  e.  M  /\  G  e.  M  /\  H  e.  M
) )  ->  (
( ( D `  H )  =  ( C `  G )  /\  ( D `  G )  =  ( C `  F ) )  ->  ( H R ( G R F ) )  =  ( ( H R G ) R F ) ) )

Proof of Theorem cmpasso
Dummy variables  f 
g  h  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmpasso.2 . . . 4  |-  D  =  ( dom_ `  T
)
2 cmpasso.5 . . . 4  |-  C  =  ( cod_ `  T
)
3 eqid 2283 . . . 4  |-  ( id_ `  T )  =  ( id_ `  T )
4 cmpasso.6 . . . 4  |-  R  =  ( o_ `  T
)
5 cmpasso.1 . . . 4  |-  M  =  dom  D
6 eqid 2283 . . . 4  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
71, 2, 3, 4, 5, 6cati 25167 . . 3  |-  ( T  e.  Cat OLD  ->  ( ( <. <. D ,  C >. ,  <. ( id_ `  T
) ,  R >. >.  e.  Ded  /\  A. f  e.  M  A. g  e.  M  A. h  e.  M  ( (
( D `  h
)  =  ( C `
 g )  /\  ( D `  g )  =  ( C `  f ) )  -> 
( h R ( g R f ) )  =  ( ( h R g ) R f ) ) )  /\  ( A. x  e.  dom  ( id_ `  T ) A. f  e.  M  ( ( C `  f )  =  x  ->  ( ( ( id_ `  T
) `  x ) R f )  =  f )  /\  A. x  e.  dom  ( id_ `  T ) A. f  e.  M  ( ( D `  f )  =  x  ->  ( f R ( ( id_ `  T ) `  x
) )  =  f ) ) ) )
8 fveq2 5525 . . . . . . . . 9  |-  ( f  =  F  ->  ( C `  f )  =  ( C `  F ) )
98eqeq2d 2294 . . . . . . . 8  |-  ( f  =  F  ->  (
( D `  g
)  =  ( C `
 f )  <->  ( D `  g )  =  ( C `  F ) ) )
109anbi2d 684 . . . . . . 7  |-  ( f  =  F  ->  (
( ( D `  h )  =  ( C `  g )  /\  ( D `  g )  =  ( C `  f ) )  <->  ( ( D `
 h )  =  ( C `  g
)  /\  ( D `  g )  =  ( C `  F ) ) ) )
11 oveq2 5866 . . . . . . . . 9  |-  ( f  =  F  ->  (
g R f )  =  ( g R F ) )
1211oveq2d 5874 . . . . . . . 8  |-  ( f  =  F  ->  (
h R ( g R f ) )  =  ( h R ( g R F ) ) )
13 oveq2 5866 . . . . . . . 8  |-  ( f  =  F  ->  (
( h R g ) R f )  =  ( ( h R g ) R F ) )
1412, 13eqeq12d 2297 . . . . . . 7  |-  ( f  =  F  ->  (
( h R ( g R f ) )  =  ( ( h R g ) R f )  <->  ( h R ( g R F ) )  =  ( ( h R g ) R F ) ) )
1510, 14imbi12d 311 . . . . . 6  |-  ( f  =  F  ->  (
( ( ( D `
 h )  =  ( C `  g
)  /\  ( D `  g )  =  ( C `  f ) )  ->  ( h R ( g R f ) )  =  ( ( h R g ) R f ) )  <->  ( (
( D `  h
)  =  ( C `
 g )  /\  ( D `  g )  =  ( C `  F ) )  -> 
( h R ( g R F ) )  =  ( ( h R g ) R F ) ) ) )
16 fveq2 5525 . . . . . . . . 9  |-  ( g  =  G  ->  ( C `  g )  =  ( C `  G ) )
1716eqeq2d 2294 . . . . . . . 8  |-  ( g  =  G  ->  (
( D `  h
)  =  ( C `
 g )  <->  ( D `  h )  =  ( C `  G ) ) )
18 fveq2 5525 . . . . . . . . 9  |-  ( g  =  G  ->  ( D `  g )  =  ( D `  G ) )
1918eqeq1d 2291 . . . . . . . 8  |-  ( g  =  G  ->  (
( D `  g
)  =  ( C `
 F )  <->  ( D `  G )  =  ( C `  F ) ) )
2017, 19anbi12d 691 . . . . . . 7  |-  ( g  =  G  ->  (
( ( D `  h )  =  ( C `  g )  /\  ( D `  g )  =  ( C `  F ) )  <->  ( ( D `
 h )  =  ( C `  G
)  /\  ( D `  G )  =  ( C `  F ) ) ) )
21 oveq1 5865 . . . . . . . . 9  |-  ( g  =  G  ->  (
g R F )  =  ( G R F ) )
2221oveq2d 5874 . . . . . . . 8  |-  ( g  =  G  ->  (
h R ( g R F ) )  =  ( h R ( G R F ) ) )
23 oveq2 5866 . . . . . . . . 9  |-  ( g  =  G  ->  (
h R g )  =  ( h R G ) )
2423oveq1d 5873 . . . . . . . 8  |-  ( g  =  G  ->  (
( h R g ) R F )  =  ( ( h R G ) R F ) )
2522, 24eqeq12d 2297 . . . . . . 7  |-  ( g  =  G  ->  (
( h R ( g R F ) )  =  ( ( h R g ) R F )  <->  ( h R ( G R F ) )  =  ( ( h R G ) R F ) ) )
2620, 25imbi12d 311 . . . . . 6  |-  ( g  =  G  ->  (
( ( ( D `
 h )  =  ( C `  g
)  /\  ( D `  g )  =  ( C `  F ) )  ->  ( h R ( g R F ) )  =  ( ( h R g ) R F ) )  <->  ( (
( D `  h
)  =  ( C `
 G )  /\  ( D `  G )  =  ( C `  F ) )  -> 
( h R ( G R F ) )  =  ( ( h R G ) R F ) ) ) )
27 fveq2 5525 . . . . . . . . 9  |-  ( h  =  H  ->  ( D `  h )  =  ( D `  H ) )
2827eqeq1d 2291 . . . . . . . 8  |-  ( h  =  H  ->  (
( D `  h
)  =  ( C `
 G )  <->  ( D `  H )  =  ( C `  G ) ) )
2928anbi1d 685 . . . . . . 7  |-  ( h  =  H  ->  (
( ( D `  h )  =  ( C `  G )  /\  ( D `  G )  =  ( C `  F ) )  <->  ( ( D `
 H )  =  ( C `  G
)  /\  ( D `  G )  =  ( C `  F ) ) ) )
30 oveq1 5865 . . . . . . . 8  |-  ( h  =  H  ->  (
h R ( G R F ) )  =  ( H R ( G R F ) ) )
31 oveq1 5865 . . . . . . . . 9  |-  ( h  =  H  ->  (
h R G )  =  ( H R G ) )
3231oveq1d 5873 . . . . . . . 8  |-  ( h  =  H  ->  (
( h R G ) R F )  =  ( ( H R G ) R F ) )
3330, 32eqeq12d 2297 . . . . . . 7  |-  ( h  =  H  ->  (
( h R ( G R F ) )  =  ( ( h R G ) R F )  <->  ( H R ( G R F ) )  =  ( ( H R G ) R F ) ) )
3429, 33imbi12d 311 . . . . . 6  |-  ( h  =  H  ->  (
( ( ( D `
 h )  =  ( C `  G
)  /\  ( D `  G )  =  ( C `  F ) )  ->  ( h R ( G R F ) )  =  ( ( h R G ) R F ) )  <->  ( (
( D `  H
)  =  ( C `
 G )  /\  ( D `  G )  =  ( C `  F ) )  -> 
( H R ( G R F ) )  =  ( ( H R G ) R F ) ) ) )
3515, 26, 34rspc3v 2893 . . . . 5  |-  ( ( F  e.  M  /\  G  e.  M  /\  H  e.  M )  ->  ( A. f  e.  M  A. g  e.  M  A. h  e.  M  ( ( ( D `  h )  =  ( C `  g )  /\  ( D `  g )  =  ( C `  f ) )  -> 
( h R ( g R f ) )  =  ( ( h R g ) R f ) )  ->  ( ( ( D `  H )  =  ( C `  G )  /\  ( D `  G )  =  ( C `  F ) )  -> 
( H R ( G R F ) )  =  ( ( H R G ) R F ) ) ) )
3635com12 27 . . . 4  |-  ( A. f  e.  M  A. g  e.  M  A. h  e.  M  (
( ( D `  h )  =  ( C `  g )  /\  ( D `  g )  =  ( C `  f ) )  ->  ( h R ( g R f ) )  =  ( ( h R g ) R f ) )  ->  (
( F  e.  M  /\  G  e.  M  /\  H  e.  M
)  ->  ( (
( D `  H
)  =  ( C `
 G )  /\  ( D `  G )  =  ( C `  F ) )  -> 
( H R ( G R F ) )  =  ( ( H R G ) R F ) ) ) )
3736ad2antlr 707 . . 3  |-  ( ( ( <. <. D ,  C >. ,  <. ( id_ `  T
) ,  R >. >.  e.  Ded  /\  A. f  e.  M  A. g  e.  M  A. h  e.  M  ( (
( D `  h
)  =  ( C `
 g )  /\  ( D `  g )  =  ( C `  f ) )  -> 
( h R ( g R f ) )  =  ( ( h R g ) R f ) ) )  /\  ( A. x  e.  dom  ( id_ `  T ) A. f  e.  M  ( ( C `  f )  =  x  ->  ( ( ( id_ `  T
) `  x ) R f )  =  f )  /\  A. x  e.  dom  ( id_ `  T ) A. f  e.  M  ( ( D `  f )  =  x  ->  ( f R ( ( id_ `  T ) `  x
) )  =  f ) ) )  -> 
( ( F  e.  M  /\  G  e.  M  /\  H  e.  M )  ->  (
( ( D `  H )  =  ( C `  G )  /\  ( D `  G )  =  ( C `  F ) )  ->  ( H R ( G R F ) )  =  ( ( H R G ) R F ) ) ) )
387, 37syl 15 . 2  |-  ( T  e.  Cat OLD  ->  ( ( F  e.  M  /\  G  e.  M  /\  H  e.  M
)  ->  ( (
( D `  H
)  =  ( C `
 G )  /\  ( D `  G )  =  ( C `  F ) )  -> 
( H R ( G R F ) )  =  ( ( H R G ) R F ) ) ) )
3938imp 418 1  |-  ( ( T  e.  Cat OLD  /\  ( F  e.  M  /\  G  e.  M  /\  H  e.  M
) )  ->  (
( ( D `  H )  =  ( C `  G )  /\  ( D `  G )  =  ( C `  F ) )  ->  ( H R ( G R F ) )  =  ( ( H R G ) R F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   dom cdm 4689   ` cfv 5255  (class class class)co 5858   dom_cdom_ 25124   cod_ccod_ 25125   id_cid_ 25126   o_co_ 25127   Dedcded 25146    Cat
OLD ccatOLD 25164
This theorem is referenced by:  dualcat2  25196  cmpassoh  25213
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-pow 4188  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-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-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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-dom_ 25129  df-cod_ 25130  df-id_ 25131  df-cmpa 25132  df-catOLD 25165
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