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Theorem evlfval 14269
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 14265 . . . 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 733 . . . . . 6  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
c  =  C )
5 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
d  =  D )
64, 5oveq12d 6058 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c  Func  d
)  =  ( C 
Func  D ) )
74fveq2d 5691 . . . . . 6  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( Base `  c )  =  ( Base `  C
) )
8 evlfval.b . . . . . 6  |-  B  =  ( Base `  C
)
97, 8syl6eqr 2454 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( Base `  c )  =  B )
10 eqidd 2405 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  f ) `  x
) )
116, 9, 10mpt2eq123dv 6095 . . . 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 4862 . . . . 5  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( ( c  Func  d )  X.  ( Base `  c ) )  =  ( ( C  Func  D )  X.  B ) )
134, 5oveq12d 6058 . . . . . . . . . 10  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c Nat  d )  =  ( C Nat  D
) )
14 evlfval.n . . . . . . . . . 10  |-  N  =  ( C Nat  D )
1513, 14syl6eqr 2454 . . . . . . . . 9  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( c Nat  d )  =  N )
1615oveqd 6057 . . . . . . . 8  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( m ( c Nat  d ) n )  =  ( m N n ) )
174fveq2d 5691 . . . . . . . . . 10  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
(  Hom  `  c )  =  (  Hom  `  C
) )
18 evlfval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  C
)
1917, 18syl6eqr 2454 . . . . . . . . 9  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
(  Hom  `  c )  =  H )
2019oveqd 6057 . . . . . . . 8  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
( ( 2nd `  x
) (  Hom  `  c
) ( 2nd `  y
) )  =  ( ( 2nd `  x
) H ( 2nd `  y ) ) )
215fveq2d 5691 . . . . . . . . . . 11  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
(comp `  d )  =  (comp `  D )
)
22 evlfval.o . . . . . . . . . . 11  |-  .x.  =  (comp `  D )
2321, 22syl6eqr 2454 . . . . . . . . . 10  |-  ( (
ph  /\  ( c  =  C  /\  d  =  D ) )  -> 
(comp `  d )  =  .x.  )
2423oveqd 6057 . . . . . . . . 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 6057 . . . . . . . 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 6095 . . . . . . 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 3236 . . . . . 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 3236 . . . . 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 6095 . . . 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 3952 . . 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 4387 . . . 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 6160 . 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 2448 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 359    = wceq 1649    e. wcel 1721   _Vcvv 2916   [_csb 3211   <.cop 3777    X. cxp 4835   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844    Func cfunc 14006   Nat cnat 14093   evalF cevlf 14261
This theorem is referenced by:  evlf2  14270  evlf1  14272  evlfcl  14274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-evlf 14265
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