MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fuccocl Structured version   Unicode version

Theorem fuccocl 15579
Description: The composition of two natural transformations is a natural transformation. Remark 6.14(a) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fuccocl.q  |-  Q  =  ( C FuncCat  D )
fuccocl.n  |-  N  =  ( C Nat  D )
fuccocl.x  |-  .xb  =  (comp `  Q )
fuccocl.r  |-  ( ph  ->  R  e.  ( F N G ) )
fuccocl.s  |-  ( ph  ->  S  e.  ( G N H ) )
Assertion
Ref Expression
fuccocl  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  e.  ( F N H ) )

Proof of Theorem fuccocl
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fuccocl.q . . . 4  |-  Q  =  ( C FuncCat  D )
2 fuccocl.n . . . 4  |-  N  =  ( C Nat  D )
3 eqid 2404 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2404 . . . 4  |-  (comp `  D )  =  (comp `  D )
5 fuccocl.x . . . 4  |-  .xb  =  (comp `  Q )
6 fuccocl.r . . . 4  |-  ( ph  ->  R  e.  ( F N G ) )
7 fuccocl.s . . . 4  |-  ( ph  ->  S  e.  ( G N H ) )
81, 2, 3, 4, 5, 6, 7fucco 15577 . . 3  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  =  ( x  e.  ( Base `  C
)  |->  ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) ) ) )
9 eqid 2404 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 eqid 2404 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
112natrcl 15565 . . . . . . . . . . 11  |-  ( R  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
126, 11syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
1312simpld 459 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
14 funcrcl 15478 . . . . . . . . 9  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1513, 14syl 17 . . . . . . . 8  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1615simprd 463 . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
1716adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
18 relfunc 15477 . . . . . . . . 9  |-  Rel  ( C  Func  D )
19 1st2ndbr 6835 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2018, 13, 19sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
213, 9, 20funcf1 15481 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
2221ffvelrnda 6011 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
232natrcl 15565 . . . . . . . . . . 11  |-  ( S  e.  ( G N H )  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D
) ) )
247, 23syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D ) ) )
2524simpld 459 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
26 1st2ndbr 6835 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
2718, 25, 26sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
283, 9, 27funcf1 15481 . . . . . . 7  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
2928ffvelrnda 6011 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  D )
)
3024simprd 463 . . . . . . . . 9  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
31 1st2ndbr 6835 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  H  e.  ( C  Func  D
) )  ->  ( 1st `  H ) ( C  Func  D )
( 2nd `  H
) )
3218, 30, 31sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1st `  H
) ( C  Func  D ) ( 2nd `  H
) )
333, 9, 32funcf1 15481 . . . . . . 7  |-  ( ph  ->  ( 1st `  H
) : ( Base `  C ) --> ( Base `  D ) )
3433ffvelrnda 6011 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  H ) `  x )  e.  (
Base `  D )
)
352, 6nat1st2nd 15566 . . . . . . . 8  |-  ( ph  ->  R  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
3635adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  R  e.  ( <. ( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
37 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
382, 36, 3, 10, 37natcl 15568 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( R `  x )  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  G
) `  x )
) )
392, 7nat1st2nd 15566 . . . . . . . 8  |-  ( ph  ->  S  e.  ( <.
( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
4039adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  S  e.  ( <. ( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
412, 40, 3, 10, 37natcl 15568 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( S `  x )  e.  ( ( ( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  H
) `  x )
) )
429, 10, 4, 17, 22, 29, 34, 38, 41catcocl 15301 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( S `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) )  e.  ( ( ( 1st `  F ) `  x
) ( Hom  `  D
) ( ( 1st `  H ) `  x
) ) )
4342ralrimiva 2820 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  C )
( ( S `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) )  e.  ( ( ( 1st `  F ) `
 x ) ( Hom  `  D )
( ( 1st `  H
) `  x )
) )
44 fvex 5861 . . . . 5  |-  ( Base `  C )  e.  _V
45 mptelixpg 7546 . . . . 5  |-  ( (
Base `  C )  e.  _V  ->  ( (
x  e.  ( Base `  C )  |->  ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) ) )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) ( Hom  `  D
) ( ( 1st `  H ) `  x
) )  <->  A. x  e.  ( Base `  C
) ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) )  e.  ( ( ( 1st `  F ) `
 x ) ( Hom  `  D )
( ( 1st `  H
) `  x )
) ) )
4644, 45ax-mp 5 . . . 4  |-  ( ( x  e.  ( Base `  C )  |->  ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) ) )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) ( Hom  `  D
) ( ( 1st `  H ) `  x
) )  <->  A. x  e.  ( Base `  C
) ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) )  e.  ( ( ( 1st `  F ) `
 x ) ( Hom  `  D )
( ( 1st `  H
) `  x )
) )
4743, 46sylibr 214 . . 3  |-  ( ph  ->  ( x  e.  (
Base `  C )  |->  ( ( S `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) ) )  e.  X_ x  e.  ( Base `  C
) ( ( ( 1st `  F ) `
 x ) ( Hom  `  D )
( ( 1st `  H
) `  x )
) )
488, 47eqeltrd 2492 . 2  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) ( Hom  `  D
) ( ( 1st `  H ) `  x
) ) )
4916adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  D  e.  Cat )
5021adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( 1st `  F ) : (
Base `  C ) --> ( Base `  D )
)
51 simpr1 1005 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  x  e.  ( Base `  C )
)
5250, 51ffvelrnd 6012 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
53 simpr2 1006 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  y  e.  ( Base `  C )
)
5450, 53ffvelrnd 6012 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
5528adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( 1st `  G ) : (
Base `  C ) --> ( Base `  D )
)
5655, 53ffvelrnd 6012 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  G ) `  y )  e.  (
Base `  D )
)
57 eqid 2404 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
5820adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( 1st `  F ) ( C 
Func  D ) ( 2nd `  F ) )
593, 57, 10, 58, 51, 53funcf2 15483 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( x
( 2nd `  F
) y ) : ( x ( Hom  `  C ) y ) --> ( ( ( 1st `  F ) `  x
) ( Hom  `  D
) ( ( 1st `  F ) `  y
) ) )
60 simpr3 1007 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  f  e.  ( x ( Hom  `  C ) y ) )
6159, 60ffvelrnd 6012 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
x ( 2nd `  F
) y ) `  f )  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) )
6235adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  R  e.  ( <. ( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
632, 62, 3, 10, 53natcl 15568 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( R `  y )  e.  ( ( ( 1st `  F
) `  y )
( Hom  `  D ) ( ( 1st `  G
) `  y )
) )
6433adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( 1st `  H ) : (
Base `  C ) --> ( Base `  D )
)
6564, 53ffvelrnd 6012 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  H ) `  y )  e.  (
Base `  D )
)
6639adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  S  e.  ( <. ( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
672, 66, 3, 10, 53natcl 15568 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( S `  y )  e.  ( ( ( 1st `  G
) `  y )
( Hom  `  D ) ( ( 1st `  H
) `  y )
) )
689, 10, 4, 49, 52, 54, 56, 61, 63, 65, 67catass 15302 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
( S `  y
) ( <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 y ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( S `  y ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( R `  y ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) ) ) )
692, 62, 3, 57, 4, 51, 53, 60nati 15570 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( R `  y )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  G
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( R `
 x ) ) )
7069oveq2d 6296 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( S `  y )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( R `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )  =  ( ( S `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( ( x ( 2nd `  G ) y ) `
 f ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( R `
 x ) ) ) )
7155, 51ffvelrnd 6012 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  D )
)
722, 62, 3, 10, 51natcl 15568 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( R `  x )  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  G
) `  x )
) )
7327adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( 1st `  G ) ( C 
Func  D ) ( 2nd `  G ) )
743, 57, 10, 73, 51, 53funcf2 15483 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( x
( 2nd `  G
) y ) : ( x ( Hom  `  C ) y ) --> ( ( ( 1st `  G ) `  x
) ( Hom  `  D
) ( ( 1st `  G ) `  y
) ) )
7574, 60ffvelrnd 6012 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
x ( 2nd `  G
) y ) `  f )  e.  ( ( ( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  G
) `  y )
) )
769, 10, 4, 49, 52, 71, 56, 72, 75, 65, 67catass 15302 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
( S `  y
) ( <. (
( 1st `  G
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( x ( 2nd `  G
) y ) `  f ) ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 x ) )  =  ( ( S `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( ( x ( 2nd `  G ) y ) `
 f ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( R `
 x ) ) ) )
772, 66, 3, 57, 4, 51, 53, 60nati 15570 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( S `  y )
( <. ( ( 1st `  G ) `  x
) ,  ( ( 1st `  G ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( x ( 2nd `  G
) y ) `  f ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  G
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( S `
 x ) ) )
7877oveq1d 6295 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
( S `  y
) ( <. (
( 1st `  G
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( x ( 2nd `  G
) y ) `  f ) ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 x ) )  =  ( ( ( ( x ( 2nd `  H ) y ) `
 f ) (
<. ( ( 1st `  G
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( S `
 x ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( R `  x ) ) )
7970, 76, 783eqtr2d 2451 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( S `  y )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( R `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )  =  ( ( ( ( x ( 2nd `  H ) y ) `
 f ) (
<. ( ( 1st `  G
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( S `
 x ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( R `  x ) ) )
8064, 51ffvelrnd 6012 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  H ) `  x )  e.  (
Base `  D )
)
812, 66, 3, 10, 51natcl 15568 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( S `  x )  e.  ( ( ( 1st `  G
) `  x )
( Hom  `  D ) ( ( 1st `  H
) `  x )
) )
8232adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( 1st `  H ) ( C 
Func  D ) ( 2nd `  H ) )
833, 57, 10, 82, 51, 53funcf2 15483 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( x
( 2nd `  H
) y ) : ( x ( Hom  `  C ) y ) --> ( ( ( 1st `  H ) `  x
) ( Hom  `  D
) ( ( 1st `  H ) `  y
) ) )
8483, 60ffvelrnd 6012 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
x ( 2nd `  H
) y ) `  f )  e.  ( ( ( 1st `  H
) `  x )
( Hom  `  D ) ( ( 1st `  H
) `  y )
) )
859, 10, 4, 49, 52, 71, 80, 72, 81, 65, 84catass 15302 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
( ( x ( 2nd `  H ) y ) `  f
) ( <. (
( 1st `  G
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( S `
 x ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( R `  x ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) ) ) )
8668, 79, 853eqtrd 2449 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
( S `  y
) ( <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 y ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) ) ) )
876adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  R  e.  ( F N G ) )
887adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  S  e.  ( G N H ) )
891, 2, 3, 4, 5, 87, 88, 53fuccoval 15578 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( S ( <. F ,  G >.  .xb  H ) R ) `  y )  =  ( ( S `
 y ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 y ) ) )
9089oveq1d 6295 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
( S ( <. F ,  G >.  .xb 
H ) R ) `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( S `
 y ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 y ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) ) )
911, 2, 3, 4, 5, 87, 88, 51fuccoval 15578 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( S ( <. F ,  G >.  .xb  H ) R ) `  x )  =  ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) ) )
9291oveq2d 6296 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
( x ( 2nd `  H ) y ) `
 f ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S ( <. F ,  G >.  .xb  H ) R ) `  x ) )  =  ( ( ( x ( 2nd `  H ) y ) `
 f ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) ) ) )
9386, 90, 923eqtr4d 2455 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( (
( S ( <. F ,  G >.  .xb 
H ) R ) `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S ( <. F ,  G >.  .xb  H ) R ) `  x ) ) )
9493ralrimivvva 2828 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) A. f  e.  ( x
( Hom  `  C ) y ) ( ( ( S ( <. F ,  G >.  .xb 
H ) R ) `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S ( <. F ,  G >.  .xb  H ) R ) `  x ) ) )
952, 3, 57, 10, 4, 13, 30isnat2 15563 . 2  |-  ( ph  ->  ( ( S (
<. F ,  G >.  .xb 
H ) R )  e.  ( F N H )  <->  ( ( S ( <. F ,  G >.  .xb  H ) R )  e.  X_ x  e.  ( Base `  C
) ( ( ( 1st `  F ) `
 x ) ( Hom  `  D )
( ( 1st `  H
) `  x )
)  /\  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) A. f  e.  ( x ( Hom  `  C ) y ) ( ( ( S ( <. F ,  G >. 
.xb  H ) R ) `  y ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S ( <. F ,  G >.  .xb  H ) R ) `  x ) ) ) ) )
9648, 94, 95mpbir2and 925 1  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  e.  ( F N H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844   A.wral 2756   _Vcvv 3061   <.cop 3980   class class class wbr 4397    |-> cmpt 4455   Rel wrel 4830   -->wf 5567   ` cfv 5571  (class class class)co 6280   1stc1st 6784   2ndc2nd 6785   X_cixp 7509   Basecbs 14843   Hom chom 14922  compcco 14923   Catccat 15280    Func cfunc 15469   Nat cnat 15556   FuncCat cfuc 15557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-fz 11729  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-hom 14935  df-cco 14936  df-cat 15284  df-func 15473  df-nat 15558  df-fuc 15559
This theorem is referenced by:  fucass  15583  fuccatid  15584  evlfcllem  15816  yonedalem3b  15874
  Copyright terms: Public domain W3C validator