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Theorem evlf2val 14092
Description: Value of the evaluation natural transformation at an object. (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 )
evlf2.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
evlf2.g  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
evlf2.x  |-  ( ph  ->  X  e.  B )
evlf2.y  |-  ( ph  ->  Y  e.  B )
evlf2.l  |-  L  =  ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. )
evlf2val.a  |-  ( ph  ->  A  e.  ( F N G ) )
evlf2val.k  |-  ( ph  ->  K  e.  ( X H Y ) )
Assertion
Ref Expression
evlf2val  |-  ( ph  ->  ( A L K )  =  ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) ) )

Proof of Theorem evlf2val
Dummy variables  a 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlfval.e . . 3  |-  E  =  ( C evalF  D )
2 evlfval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 evlfval.d . . 3  |-  ( ph  ->  D  e.  Cat )
4 evlfval.b . . 3  |-  B  =  ( Base `  C
)
5 evlfval.h . . 3  |-  H  =  (  Hom  `  C
)
6 evlfval.o . . 3  |-  .x.  =  (comp `  D )
7 evlfval.n . . 3  |-  N  =  ( C Nat  D )
8 evlf2.f . . 3  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
9 evlf2.g . . 3  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
10 evlf2.x . . 3  |-  ( ph  ->  X  e.  B )
11 evlf2.y . . 3  |-  ( ph  ->  Y  e.  B )
12 evlf2.l . . 3  |-  L  =  ( <. F ,  X >. ( 2nd `  E
) <. G ,  Y >. )
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12evlf2 14091 . 2  |-  ( ph  ->  L  =  ( a  e.  ( F N G ) ,  g  e.  ( X H Y )  |->  ( ( a `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  g ) ) ) )
14 simprl 732 . . . 4  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
a  =  A )
1514fveq1d 5610 . . 3  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
( a `  Y
)  =  ( A `
 Y ) )
16 simprr 733 . . . 4  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
g  =  K )
1716fveq2d 5612 . . 3  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
( ( X ( 2nd `  F ) Y ) `  g
)  =  ( ( X ( 2nd `  F
) Y ) `  K ) )
1815, 17oveq12d 5963 . 2  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
( ( a `  Y ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >.  .x.  ( ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
 g ) )  =  ( ( A `
 Y ) (
<. ( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >.  .x.  ( ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
 K ) ) )
19 evlf2val.a . 2  |-  ( ph  ->  A  e.  ( F N G ) )
20 evlf2val.k . 2  |-  ( ph  ->  K  e.  ( X H Y ) )
21 ovex 5970 . . 3  |-  ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) )  e. 
_V
2221a1i 10 . 2  |-  ( ph  ->  ( ( A `  Y ) ( <.
( ( 1st `  F
) `  X ) ,  ( ( 1st `  F ) `  Y
) >.  .x.  ( ( 1st `  G ) `  Y ) ) ( ( X ( 2nd `  F ) Y ) `
 K ) )  e.  _V )
2313, 18, 19, 20, 22ovmpt2d 6062 1  |-  ( ph  ->  ( A L K )  =  ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   <.cop 3719   ` cfv 5337  (class class class)co 5945   1stc1st 6207   2ndc2nd 6208   Basecbs 13245    Hom chom 13316  compcco 13317   Catccat 13665    Func cfunc 13827   Nat cnat 13914   evalF cevlf 14082
This theorem is referenced by:  evlfcllem  14094  evlfcl  14095  uncf2  14110  yonedalem3b  14152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-evlf 14086
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