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Theorem cmpasso 25876
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 2296 . . . 4  |-  ( id_ `  T )  =  ( id_ `  T )
4 cmpasso.6 . . . 4  |-  R  =  ( o_ `  T
)
5 cmpasso.1 . . . 4  |-  M  =  dom  D
6 eqid 2296 . . . 4  |-  dom  ( id_ `  T )  =  dom  ( id_ `  T
)
71, 2, 3, 4, 5, 6cati 25858 . . 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 5541 . . . . . . . . 9  |-  ( f  =  F  ->  ( C `  f )  =  ( C `  F ) )
98eqeq2d 2307 . . . . . . . 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 5882 . . . . . . . . 9  |-  ( f  =  F  ->  (
g R f )  =  ( g R F ) )
1211oveq2d 5890 . . . . . . . 8  |-  ( f  =  F  ->  (
h R ( g R f ) )  =  ( h R ( g R F ) ) )
13 oveq2 5882 . . . . . . . 8  |-  ( f  =  F  ->  (
( h R g ) R f )  =  ( ( h R g ) R F ) )
1412, 13eqeq12d 2310 . . . . . . 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 5541 . . . . . . . . 9  |-  ( g  =  G  ->  ( C `  g )  =  ( C `  G ) )
1716eqeq2d 2307 . . . . . . . 8  |-  ( g  =  G  ->  (
( D `  h
)  =  ( C `
 g )  <->  ( D `  h )  =  ( C `  G ) ) )
18 fveq2 5541 . . . . . . . . 9  |-  ( g  =  G  ->  ( D `  g )  =  ( D `  G ) )
1918eqeq1d 2304 . . . . . . . 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 5881 . . . . . . . . 9  |-  ( g  =  G  ->  (
g R F )  =  ( G R F ) )
2221oveq2d 5890 . . . . . . . 8  |-  ( g  =  G  ->  (
h R ( g R F ) )  =  ( h R ( G R F ) ) )
23 oveq2 5882 . . . . . . . . 9  |-  ( g  =  G  ->  (
h R g )  =  ( h R G ) )
2423oveq1d 5889 . . . . . . . 8  |-  ( g  =  G  ->  (
( h R g ) R F )  =  ( ( h R G ) R F ) )
2522, 24eqeq12d 2310 . . . . . . 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 5541 . . . . . . . . 9  |-  ( h  =  H  ->  ( D `  h )  =  ( D `  H ) )
2827eqeq1d 2304 . . . . . . . 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 5881 . . . . . . . 8  |-  ( h  =  H  ->  (
h R ( G R F ) )  =  ( H R ( G R F ) ) )
31 oveq1 5881 . . . . . . . . 9  |-  ( h  =  H  ->  (
h R G )  =  ( H R G ) )
3231oveq1d 5889 . . . . . . . 8  |-  ( h  =  H  ->  (
( h R G ) R F )  =  ( ( H R G ) R F ) )
3330, 32eqeq12d 2310 . . . . . . 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 2906 . . . . 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 1632    e. wcel 1696   A.wral 2556   <.cop 3656   dom cdm 4705   ` cfv 5271  (class class class)co 5874   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   o_co_ 25818   Dedcded 25837    Cat
OLD ccatOLD 25855
This theorem is referenced by:  dualcat2  25887  cmpassoh  25904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-catOLD 25856
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