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Theorem evlfval 14319
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 14315 . . . 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 734 . . . . . 6  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
c  =  C )
5 simprr 735 . . . . . 6  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
d  =  D )
64, 5oveq12d 6102 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c  Func  d
)  =  ( C 
Func  D ) )
74fveq2d 5735 . . . . . 6  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( Base `  c )  =  ( Base `  C
) )
8 evlfval.b . . . . . 6  |-  B  =  ( Base `  C
)
97, 8syl6eqr 2488 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( Base `  c )  =  B )
10 eqidd 2439 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  f ) `  x
) )
116, 9, 10mpt2eq123dv 6139 . . . 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 4906 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( ( c  Func  d )  X.  ( Base `  c ) )  =  ( ( C  Func  D )  X.  B ) )
134, 5oveq12d 6102 . . . . . . . . . 10  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c Nat  d )  =  ( C Nat  D
) )
14 evlfval.n . . . . . . . . . 10  |-  N  =  ( C Nat  D )
1513, 14syl6eqr 2488 . . . . . . . . 9  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c Nat  d )  =  N )
1615oveqd 6101 . . . . . . . 8  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( m ( c Nat  d ) n )  =  ( m N n ) )
174fveq2d 5735 . . . . . . . . . 10  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
(  Hom  `  c )  =  (  Hom  `  C
) )
18 evlfval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  C
)
1917, 18syl6eqr 2488 . . . . . . . . 9  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
(  Hom  `  c )  =  H )
2019oveqd 6101 . . . . . . . 8  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( ( 2nd `  x
) (  Hom  `  c
) ( 2nd `  y
) )  =  ( ( 2nd `  x
) H ( 2nd `  y ) ) )
215fveq2d 5735 . . . . . . . . . . 11  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
(comp `  d )  =  (comp `  D )
)
22 evlfval.o . . . . . . . . . . 11  |-  .x.  =  (comp `  D )
2321, 22syl6eqr 2488 . . . . . . . . . 10  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
(comp `  d )  =  .x.  )
2423oveqd 6101 . . . . . . . . 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 6101 . . . . . . . 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 6139 . . . . . . 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 3278 . . . . . 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 3278 . . . . 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 6139 . . . 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 3994 . . 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 4430 . . . 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 6204 . 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 2482 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   [_csb 3253   <.cop 3819    X. cxp 4879   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894    Func cfunc 14056   Nat cnat 14143   evalF cevlf 14311
This theorem is referenced by:  evlf2  14320  evlf1  14322  evlfcl  14324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-evlf 14315
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