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Theorem evlfval 15340
Description: Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlfval.e  |-  E  =  ( C evalF  D )
evlfval.c  |-  ( ph  ->  C  e.  Cat )
evlfval.d  |-  ( ph  ->  D  e.  Cat )
evlfval.b  |-  B  =  ( Base `  C
)
evlfval.h  |-  H  =  ( Hom  `  C
)
evlfval.o  |-  .x.  =  (comp `  D )
evlfval.n  |-  N  =  ( C Nat  D )
Assertion
Ref Expression
evlfval  |-  ( ph  ->  E  =  <. (
f  e.  ( C 
Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `
 x ) ) ,  ( x  e.  ( ( C  Func  D )  X.  B ) ,  y  e.  ( ( C  Func  D
)  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
>. )
Distinct variable groups:    f, a,
g, m, n, x, y, C    D, a,
f, g, m, n, x, y    g, H, m, n, x, y    N, a, g, m, n, x, y    ph, a,
f, g, m, n, x, y    .x. , a,
g, m, n, x, y    x, B, y
Allowed substitution hints:    B( f, g, m, n, a)    .x. ( f)    E( x, y, f, g, m, n, a)    H( f, a)    N( f)

Proof of Theorem evlfval
Dummy variables  c 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlfval.e . 2  |-  E  =  ( C evalF  D )
2 df-evlf 15336 . . . 4  |- evalF  =  ( c  e. 
Cat ,  d  e.  Cat  |->  <. ( f  e.  ( c  Func  d
) ,  x  e.  ( Base `  c
)  |->  ( ( 1st `  f ) `  x
) ) ,  ( x  e.  ( ( c  Func  d )  X.  ( Base `  c
) ) ,  y  e.  ( ( c 
Func  d )  X.  ( Base `  c
) )  |->  [_ ( 1st `  x )  /  m ]_ [_ ( 1st `  y )  /  n ]_ ( a  e.  ( m ( c Nat  d
) n ) ,  g  e.  ( ( 2nd `  x ) ( Hom  `  c
) ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  d )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) ) ) >. )
32a1i 11 . . 3  |-  ( ph  -> evalF  =  ( c  e.  Cat ,  d  e.  Cat  |->  <.
( f  e.  ( c  Func  d ) ,  x  e.  ( Base `  c )  |->  ( ( 1st `  f
) `  x )
) ,  ( x  e.  ( ( c 
Func  d )  X.  ( Base `  c
) ) ,  y  e.  ( ( c 
Func  d )  X.  ( Base `  c
) )  |->  [_ ( 1st `  x )  /  m ]_ [_ ( 1st `  y )  /  n ]_ ( a  e.  ( m ( c Nat  d
) n ) ,  g  e.  ( ( 2nd `  x ) ( Hom  `  c
) ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  d )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) ) ) >. ) )
4 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
c  =  C )
5 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
d  =  D )
64, 5oveq12d 6300 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c  Func  d
)  =  ( C 
Func  D ) )
74fveq2d 5868 . . . . . 6  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( Base `  c )  =  ( Base `  C
) )
8 evlfval.b . . . . . 6  |-  B  =  ( Base `  C
)
97, 8syl6eqr 2526 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( Base `  c )  =  B )
10 eqidd 2468 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  f ) `  x
) )
116, 9, 10mpt2eq123dv 6341 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( f  e.  ( c  Func  d ) ,  x  e.  ( Base `  c )  |->  ( ( 1st `  f
) `  x )
)  =  ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `  x
) ) )
126, 9xpeq12d 5024 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( ( c  Func  d )  X.  ( Base `  c ) )  =  ( ( C  Func  D )  X.  B ) )
134, 5oveq12d 6300 . . . . . . . . . 10  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c Nat  d )  =  ( C Nat  D
) )
14 evlfval.n . . . . . . . . . 10  |-  N  =  ( C Nat  D )
1513, 14syl6eqr 2526 . . . . . . . . 9  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c Nat  d )  =  N )
1615oveqd 6299 . . . . . . . 8  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( m ( c Nat  d ) n )  =  ( m N n ) )
174fveq2d 5868 . . . . . . . . . 10  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( Hom  `  c )  =  ( Hom  `  C
) )
18 evlfval.h . . . . . . . . . 10  |-  H  =  ( Hom  `  C
)
1917, 18syl6eqr 2526 . . . . . . . . 9  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( Hom  `  c )  =  H )
2019oveqd 6299 . . . . . . . 8  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( ( 2nd `  x
) ( Hom  `  c
) ( 2nd `  y
) )  =  ( ( 2nd `  x
) H ( 2nd `  y ) ) )
215fveq2d 5868 . . . . . . . . . . 11  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
(comp `  d )  =  (comp `  D )
)
22 evlfval.o . . . . . . . . . . 11  |-  .x.  =  (comp `  D )
2321, 22syl6eqr 2526 . . . . . . . . . 10  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
(comp `  d )  =  .x.  )
2423oveqd 6299 . . . . . . . . 9  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>. (comp `  d )
( ( 1st `  n
) `  ( 2nd `  y ) ) )  =  ( <. (
( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.  .x.  ( ( 1st `  n
) `  ( 2nd `  y ) ) ) )
2524oveqd 6299 . . . . . . . 8  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( ( a `  ( 2nd `  y ) ) ( <. (
( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  d )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) )  =  ( ( a `  ( 2nd `  y ) ) ( <. (
( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.  .x.  ( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) )
2616, 20, 25mpt2eq123dv 6341 . . . . . . 7  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( a  e.  ( m ( c Nat  d
) n ) ,  g  e.  ( ( 2nd `  x ) ( Hom  `  c
) ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  d )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) )  =  ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
2726csbeq2dv 3835 . . . . . 6  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  ->  [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m ( c Nat  d
) n ) ,  g  e.  ( ( 2nd `  x ) ( Hom  `  c
) ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  d )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) )  =  [_ ( 1st `  y )  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
2827csbeq2dv 3835 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  ->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m ( c Nat  d
) n ) ,  g  e.  ( ( 2nd `  x ) ( Hom  `  c
) ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  d )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) )  =  [_ ( 1st `  x )  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
2912, 12, 28mpt2eq123dv 6341 . . . 4  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( x  e.  ( ( c  Func  d
)  X.  ( Base `  c ) ) ,  y  e.  ( ( c  Func  d )  X.  ( Base `  c
) )  |->  [_ ( 1st `  x )  /  m ]_ [_ ( 1st `  y )  /  n ]_ ( a  e.  ( m ( c Nat  d
) n ) ,  g  e.  ( ( 2nd `  x ) ( Hom  `  c
) ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  d )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) ) )  =  ( x  e.  ( ( C 
Func  D )  X.  B
) ,  y  e.  ( ( C  Func  D )  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) ) )
3011, 29opeq12d 4221 . . 3  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  ->  <. ( f  e.  ( c  Func  d ) ,  x  e.  ( Base `  c )  |->  ( ( 1st `  f
) `  x )
) ,  ( x  e.  ( ( c 
Func  d )  X.  ( Base `  c
) ) ,  y  e.  ( ( c 
Func  d )  X.  ( Base `  c
) )  |->  [_ ( 1st `  x )  /  m ]_ [_ ( 1st `  y )  /  n ]_ ( a  e.  ( m ( c Nat  d
) n ) ,  g  e.  ( ( 2nd `  x ) ( Hom  `  c
) ( 2nd `  y
) )  |->  ( ( a `  ( 2nd `  y ) ) (
<. ( ( 1st `  m
) `  ( 2nd `  x ) ) ,  ( ( 1st `  m
) `  ( 2nd `  y ) ) >.
(comp `  d )
( ( 1st `  n
) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m ) ( 2nd `  y ) ) `  g ) ) ) ) >.  =  <. ( f  e.  ( C 
Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `
 x ) ) ,  ( x  e.  ( ( C  Func  D )  X.  B ) ,  y  e.  ( ( C  Func  D
)  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
>. )
31 evlfval.c . . 3  |-  ( ph  ->  C  e.  Cat )
32 evlfval.d . . 3  |-  ( ph  ->  D  e.  Cat )
33 opex 4711 . . . 4  |-  <. (
f  e.  ( C 
Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `
 x ) ) ,  ( x  e.  ( ( C  Func  D )  X.  B ) ,  y  e.  ( ( C  Func  D
)  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
>.  e.  _V
3433a1i 11 . . 3  |-  ( ph  -> 
<. ( f  e.  ( C  Func  D ) ,  x  e.  B  |->  ( ( 1st `  f
) `  x )
) ,  ( x  e.  ( ( C 
Func  D )  X.  B
) ,  y  e.  ( ( C  Func  D )  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
>.  e.  _V )
353, 30, 31, 32, 34ovmpt2d 6412 . 2  |-  ( ph  ->  ( C evalF  D )  =  <. ( f  e.  ( C 
Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `
 x ) ) ,  ( x  e.  ( ( C  Func  D )  X.  B ) ,  y  e.  ( ( C  Func  D
)  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
>. )
361, 35syl5eq 2520 1  |-  ( ph  ->  E  =  <. (
f  e.  ( C 
Func  D ) ,  x  e.  B  |->  ( ( 1st `  f ) `
 x ) ) ,  ( x  e.  ( ( C  Func  D )  X.  B ) ,  y  e.  ( ( C  Func  D
)  X.  B ) 
|->  [_ ( 1st `  x
)  /  m ]_ [_ ( 1st `  y
)  /  n ]_ ( a  e.  ( m N n ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( ( a `  ( 2nd `  y ) ) ( <. ( ( 1st `  m ) `  ( 2nd `  x ) ) ,  ( ( 1st `  m ) `  ( 2nd `  y ) )
>.  .x.  ( ( 1st `  n ) `  ( 2nd `  y ) ) ) ( ( ( 2nd `  x ) ( 2nd `  m
) ( 2nd `  y
) ) `  g
) ) ) )
>. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   [_csb 3435   <.cop 4033    X. cxp 4997   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   Basecbs 14486   Hom chom 14562  compcco 14563   Catccat 14915    Func cfunc 15077   Nat cnat 15164   evalF cevlf 15332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-evlf 15336
This theorem is referenced by:  evlf2  15341  evlf1  15343  evlfcl  15345
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