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Theorem catass 13911
Description: Associativity of composition in a category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
catcocl.b  |-  B  =  ( Base `  C
)
catcocl.h  |-  H  =  (  Hom  `  C
)
catcocl.o  |-  .x.  =  (comp `  C )
catcocl.c  |-  ( ph  ->  C  e.  Cat )
catcocl.x  |-  ( ph  ->  X  e.  B )
catcocl.y  |-  ( ph  ->  Y  e.  B )
catcocl.z  |-  ( ph  ->  Z  e.  B )
catcocl.f  |-  ( ph  ->  F  e.  ( X H Y ) )
catcocl.g  |-  ( ph  ->  G  e.  ( Y H Z ) )
catass.w  |-  ( ph  ->  W  e.  B )
catass.g  |-  ( ph  ->  K  e.  ( Z H W ) )
Assertion
Ref Expression
catass  |-  ( ph  ->  ( ( K (
<. Y ,  Z >.  .x. 
W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) )

Proof of Theorem catass
Dummy variables  f 
g  k  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcocl.c . . 3  |-  ( ph  ->  C  e.  Cat )
2 catcocl.b . . . . 5  |-  B  =  ( Base `  C
)
3 catcocl.h . . . . 5  |-  H  =  (  Hom  `  C
)
4 catcocl.o . . . . 5  |-  .x.  =  (comp `  C )
52, 3, 4iscat 13897 . . . 4  |-  ( C  e.  Cat  ->  ( C  e.  Cat  <->  A. x  e.  B  ( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  (
y H x ) ( g ( <.
y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y
>.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) ) ) ) )
65ibi 233 . . 3  |-  ( C  e.  Cat  ->  A. x  e.  B  ( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  (
y H x ) ( g ( <.
y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y
>.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) ) ) )
71, 6syl 16 . 2  |-  ( ph  ->  A. x  e.  B  ( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  ( y H x ) ( g ( <.
y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y
>.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) ) ) )
8 catcocl.x . . 3  |-  ( ph  ->  X  e.  B )
9 catcocl.y . . . . . 6  |-  ( ph  ->  Y  e.  B )
109adantr 452 . . . . 5  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
11 catcocl.z . . . . . . 7  |-  ( ph  ->  Z  e.  B )
1211ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  Z  e.  B )
13 catcocl.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( X H Y ) )
1413ad3antrrr 711 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  F  e.  ( X H Y ) )
15 simpllr 736 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  x  =  X )
16 simplr 732 . . . . . . . . 9  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  y  =  Y )
1715, 16oveq12d 6099 . . . . . . . 8  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  (
x H y )  =  ( X H Y ) )
1814, 17eleqtrrd 2513 . . . . . . 7  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  F  e.  ( x H y ) )
19 catcocl.g . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( Y H Z ) )
2019ad4antr 713 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  G  e.  ( Y H Z ) )
21 simpllr 736 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  y  =  Y )
22 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  z  =  Z )
2321, 22oveq12d 6099 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  (
y H z )  =  ( Y H Z ) )
2420, 23eleqtrrd 2513 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  G  e.  ( y H z ) )
25 catass.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  B )
2625ad5antr 715 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  W  e.  B )
27 catass.g . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  ( Z H W ) )
2827ad6antr 717 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  K  e.  ( Z H W ) )
29 simp-4r 744 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  z  =  Z )
30 simpr 448 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  w  =  W )
3129, 30oveq12d 6099 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  (
z H w )  =  ( Z H W ) )
3228, 31eleqtrrd 2513 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  K  e.  ( z H w ) )
33 simp-7r 750 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  x  =  X )
34 simp-6r 748 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  y  =  Y )
3533, 34opeq12d 3992 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  <. x ,  y >.  =  <. X ,  Y >. )
36 simplr 732 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  w  =  W )
3735, 36oveq12d 6099 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  ( <. x ,  y >.  .x.  w )  =  (
<. X ,  Y >.  .x. 
W ) )
38 simp-5r 746 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  z  =  Z )
3934, 38opeq12d 3992 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  <. y ,  z >.  =  <. Y ,  Z >. )
4039, 36oveq12d 6099 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  ( <. y ,  z >.  .x.  w )  =  (
<. Y ,  Z >.  .x. 
W ) )
41 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  k  =  K )
42 simpllr 736 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  g  =  G )
4340, 41, 42oveq123d 6102 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  (
k ( <. y ,  z >.  .x.  w
) g )  =  ( K ( <. Y ,  Z >.  .x. 
W ) G ) )
44 simp-4r 744 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  f  =  F )
4537, 43, 44oveq123d 6102 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  (
( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F ) )
4633, 38opeq12d 3992 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  <. x ,  z >.  =  <. X ,  Z >. )
4746, 36oveq12d 6099 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  ( <. x ,  z >.  .x.  w )  =  (
<. X ,  Z >.  .x. 
W ) )
4835, 38oveq12d 6099 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  ( <. x ,  y >.  .x.  z )  =  (
<. X ,  Y >.  .x. 
Z ) )
4948, 42, 44oveq123d 6102 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  (
g ( <. x ,  y >.  .x.  z
) f )  =  ( G ( <. X ,  Y >.  .x. 
Z ) F ) )
5047, 41, 49oveq123d 6102 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  (
k ( <. x ,  z >.  .x.  w
) ( g (
<. x ,  y >.  .x.  z ) f ) )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) )
5145, 50eqeq12d 2450 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  /\  k  =  K )  ->  (
( ( k (
<. y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) )  <-> 
( ( K (
<. Y ,  Z >.  .x. 
W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5232, 51rspcdv 3055 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  /\  w  =  W )  ->  ( A. k  e.  (
z H w ) ( ( k (
<. y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W )
( G ( <. X ,  Y >.  .x. 
Z ) F ) ) ) )
5326, 52rspcimdv 3053 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  ( A. w  e.  B  A. k  e.  (
z H w ) ( ( k (
<. y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W )
( G ( <. X ,  Y >.  .x. 
Z ) F ) ) ) )
5453adantld 454 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  /\  g  =  G )  ->  (
( ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F )  =  ( K (
<. X ,  Z >.  .x. 
W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5524, 54rspcimdv 3053 . . . . . . 7  |-  ( ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  /\  z  =  Z )  /\  f  =  F )  ->  ( A. g  e.  (
y H z ) ( ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F )  =  ( K (
<. X ,  Z >.  .x. 
W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5618, 55rspcimdv 3053 . . . . . 6  |-  ( ( ( ( ph  /\  x  =  X )  /\  y  =  Y
)  /\  z  =  Z )  ->  ( A. f  e.  (
x H y ) A. g  e.  ( y H z ) ( ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F )  =  ( K (
<. X ,  Z >.  .x. 
W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5712, 56rspcimdv 3053 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( A. z  e.  B  A. f  e.  (
x H y ) A. g  e.  ( y H z ) ( ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F )  =  ( K (
<. X ,  Z >.  .x. 
W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5810, 57rspcimdv 3053 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  ( A. y  e.  B  A. z  e.  B  A. f  e.  (
x H y ) A. g  e.  ( y H z ) ( ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) )  ->  ( ( K ( <. Y ,  Z >.  .x.  W ) G ) ( <. X ,  Y >.  .x. 
W ) F )  =  ( K (
<. X ,  Z >.  .x. 
W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
5958adantld 454 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (
( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  ( y H x ) ( g (
<. y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y
>.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) ) )  ->  (
( K ( <. Y ,  Z >.  .x. 
W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
608, 59rspcimdv 3053 . 2  |-  ( ph  ->  ( A. x  e.  B  ( E. g  e.  ( x H x ) A. y  e.  B  ( A. f  e.  ( y H x ) ( g (
<. y ,  x >.  .x.  x ) f )  =  f  /\  A. f  e.  ( x H y ) ( f ( <. x ,  x >.  .x.  y ) g )  =  f )  /\  A. y  e.  B  A. z  e.  B  A. f  e.  ( x H y ) A. g  e.  ( y H z ) ( ( g ( <. x ,  y
>.  .x.  z ) f )  e.  ( x H z )  /\  A. w  e.  B  A. k  e.  ( z H w ) ( ( k ( <.
y ,  z >.  .x.  w ) g ) ( <. x ,  y
>.  .x.  w ) f )  =  ( k ( <. x ,  z
>.  .x.  w ) ( g ( <. x ,  y >.  .x.  z
) f ) ) ) )  ->  (
( K ( <. Y ,  Z >.  .x. 
W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) ) )
617, 60mpd 15 1  |-  ( ph  ->  ( ( K (
<. Y ,  Z >.  .x. 
W ) G ) ( <. X ,  Y >.  .x.  W ) F )  =  ( K ( <. X ,  Z >.  .x.  W ) ( G ( <. X ,  Y >.  .x.  Z ) F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   <.cop 3817   ` cfv 5454  (class class class)co 6081   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889
This theorem is referenced by:  oppccatid  13945  sectcan  13981  sectco  13982  sectmon  14003  monsect  14004  subccatid  14043  fuccocl  14161  fucass  14165  invfuc  14171  arwass  14229  xpccatid  14285  evlfcllem  14318  hofcllem  14355
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
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 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-cat 13893
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