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Theorem evlf2val 15048
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 15047 . 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 755 . . . 4  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
a  =  A )
1514fveq1d 5712 . . 3  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
( a `  Y
)  =  ( A `
 Y ) )
16 simprr 756 . . . 4  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
g  =  K )
1716fveq2d 5714 . . 3  |-  ( (
ph  /\  ( a  =  A  /\  g  =  K ) )  -> 
( ( X ( 2nd `  F ) Y ) `  g
)  =  ( ( X ( 2nd `  F
) Y ) `  K ) )
1815, 17oveq12d 6128 . 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 6135 . . 3  |-  ( ( A `  Y ) ( <. ( ( 1st `  F ) `  X
) ,  ( ( 1st `  F ) `
 Y ) >.  .x.  ( ( 1st `  G
) `  Y )
) ( ( X ( 2nd `  F
) Y ) `  K ) )  e. 
_V
2221a1i 11 . 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 6237 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2991   <.cop 3902   ` cfv 5437  (class class class)co 6110   1stc1st 6594   2ndc2nd 6595   Basecbs 14193   Hom chom 14268  compcco 14269   Catccat 14621    Func cfunc 14783   Nat cnat 14870   evalF cevlf 15038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-evlf 15042
This theorem is referenced by:  evlfcllem  15050  evlfcl  15051  uncf2  15066  yonedalem3b  15108
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